A model p edic i e scheduling s a egy o coo dina ed inland essel
na iga ion and b idge ope a ion
Pablo Sego ia1, Vicenc¸ Puig2and Vasso Reppa1
Abs ac — This pape p esen s he design o a model p e-
dic i e scheduling s a egy o add ess he inland wa e bo ne
anspo (IWT) p oblem conside ing b idges ha mus open o
enable essel passage. The main con ibu ion is he o mula ion
o a con ol-o ien ed model o he p oblem, including p oposi-
ional logic exp essions ha cha ac e ize sys em beha io and
hei con e sion in o (in)equali y cons ain s. The esul ing
model is embedded in o a p edic i e scheduling app oach o
de e mine b idge opening ime ables and essel passage imes
in a coo dina ed manne . The e ec i eness o he s a egy is
demons a ed on a ealis ic case s udy based on he Rhine-
Alpine co ido .
I. INTRODUCTION
F eigh anspo a ion is an essen ial p ocess wi hin he
supply chain, as i allows o ans e goods in an e icien
manne and ensu e hei imely a ailabili y a he des ina-
ion [1]. While se e al di e en anspo modes may be
used, inland wa e bo ne anspo (IWT) eme ges as a cos -
e ec i e and en i onmen ally- iendly al e na i e o mo e
la ge amoun s o ca go [2]. Despi e he ad an ages i o e s,
IWT only ep esen ed 4% o he o al goods anspo ed in
he EU-28 in 2016 [3]. This is mainly due o he ac ha
eliabili y o ope a ions is nega i ely impac ed by inaccu a e
in o ma ion a se ice le el and high sys em conges ion, and
hus calls o communica ion among in e es ed pa ies [4].
IWT encompasses he simul aneous ope a ion o es-
sels and in as uc u e in c amped a eas, which a e mo e-
o e cha ac e ized by con lic ing ope a ional objec i es, hus
ende ing IWT a challenging p oblem. While inadequa e
solu ions may lead o subop imal na iga ion and in as-
uc u e u iliza ion, he conside a ion o essel- o- essel
(V2V), in as uc u e- o-in as uc u e (I2I) and essel- o-
in as uc u e (V2I) communica ion in an agen -based ame-
wo k has he po en ial o yield imp o ed solu ions, as in-
en ions om one agen can be an icipa ed by o he s and
p epa ed o in ad ance.
Zooming in on V2I communica ion, he mos common
pieces o in as uc u e encoun e ed in wa e way ne wo ks
This wo k was suppo ed in pa by he Resea chlab Au onomous
Shipping (RAS) o Del Uni e si y o Technology, in pa by he p ojec
”No el inland wa e way anspo concep s o mo ing eigh e ec i ely
(NOVIMOVE)” ( his p ojec has ecei ed unding om he Eu opean
Union’s Ho izon 2020 esea ch and inno a ion p og amme unde g an
ag eemen No 858508), and in pa by he Spanish S a e Resea ch Agency
(AEI) and he Eu opean Regional De elopmen Fund (ERFD) h ough he
p ojec SaCoAV ( e . MINECO PID2020-114244RB-I00).
1Depa men o Ma i ime and T anspo Technology, Del Uni e -
si y o Technology, Del , he Ne he lands (e-mail: {p.sego iacas illo,
. eppa}@ udel .nl).
2Ad anced Con ol Sys ems G oup, Uni e si a Poli `
ecnica de Ca alunya,
Ins i u de Rob`
o ica i In o m`
a ica Indus ial (CSIC-UPC), Ba celona, Spain
(e-mail: [email p o ec ed]).
a e b idges—mo able and ixed—and locks. Vessels mus
pass h ough in as uc u e on hei way owa ds des ina ion,
which hey aim o do wi h minimal wai ing imes. Lock
scheduling has been widely s udied, conside ing bo h single-
chambe [5], [6] and mul iple-chambe [7], [8] se ial lock
con igu a ions. Su p isingly enough, scheduling o mo able
b idges has no ecei ed he same deg ee o a en ion despi e
he ac ha hei ope a ion has simul aneous implica ions o
oad, ailway and wa e bo ne anspo , being [9] one o he
ew pape s on he opic. Howe e , b idges a e conside ed
o be cha ac e ized by p ede ined opening egimes, and
he e o e he app oach does no u ilize peak and alley
passage demand o adjus openings.
This pape ex ends he p elimina y wo k ca ied ou in
[9], whe e he ope a ion ime ables we e ixed, by conside -
ing dynamic b idge ope a ion. In o he wo ds, b idges can
ope a e on demand and hus adap hei opening egimes o
essel passage needs, which a e p o ided ia V2I communi-
ca ion. Mo eo e , a con ol-o ien ed model o he p ocess is
designed, oge he wi h p oposi ional logic exp essions ha
go e n sys em beha io . These a e ansla ed in o (in)equali y
cons ain s and in eg a ed in o he design o a model p e-
dic i e scheduling s a egy, which de e mines passage imes
o e e y essel-b idge pai and communica es hese plans
o each o he essels. Mo eo e , he esul ing opening
schedules a e sha ed wi h he b idges.
The es o he pape is o ganized as ollows: he dynamic
b idge opening scheduling p oblem is desc ibed in Sec ion II,
and an app oach o sol e he p oblem is p esen ed in Sec-
ion III. A case s udy based on he Rhine-Alpine co ido
se es o es he e ec i eness o he app oach in Sec ion IV,
allowing o d aw conclusions and es ablish u u e esea ch
a enues in Sec ion V.
II. PROBLEM STATEMENT
The dynamic b idge opening scheduling p oblem can be
o mula ed as ollows. A se o essels Vmus pass a se o
mo able b idges Bwhile sailing om o igin o des ina ion,
wi h |B| =nand |V| =m. A disc e e p oblem se ing is
adop ed, and hus ime is di ided in o a se o ime s eps K
o equal leng h. Fu he mo e:
•B idge i,i∈ {1, ..., n}, is cha ac e ized by i s nominal
wid h, b(i)[m], he maximum numbe o consecu i e
ime s eps i can s ay open (so as o limi a ic
dis up ion on b idge deck), N(i)
up , and he minimum
numbe o consecu i e ime s eps i mus emain closed
immedia ely a e an open-close swi ch, N(i)
down,∀i∈ B.
Nup and Ndown can also be e e ed o as maximum up-
ime and minimum down- ime, espec i ely. B idges a e
numbe ed such ha i= 1 and i=nco espond o he
i s and las b idge o be passed h ough, espec i ely.
•Vessel j,j∈ {1, ..., m}, is cha ac e ized by i s wid h,
(j)[m], which includes he sa e y dis ance be ween
essel jand he es o essels, and i s oyage plans,
ep esen ed by ea lies and op imal passage ins an s
h ough b idge i,τ(i,j)
e, τ(i,j)
o∈Z+, espec i ely, ∀i∈
B,∀j∈ V, wi h Z+ he se o posi i e in ege s.
Ea lies and op imal passage ins an s can be compu ed
conside ing he maximum speed and he speed ha
minimizes uel consump ion, espec i ely, and in e -
b idge dis ances, and a e known be o e he s a o
essel jou neys.
The assump ions o he p oblem a e lis ed below:
•Vessels may en e he sys em a any ime ins an k∈ K.
Once hey ha e been scheduled h ough all b idges, hey
a e no longe aken in o conside a ion.
•Clea ance unde b idges (measu ed om wa e su ace
o b idge unde side) is no su icien o essels o sail
below b idges while hese a e closed.
•Mul iple essels may pass a b idge simul aneously
p o ided ha hei combined wid h does no exceed he
nominal wid h o he b idge.
•Choice o ime s ep size is su icien ly la ge o essel
i o pass h ough b idge jin one ime s ep wi h ze o
dwell ime, ∀i∈ B,∀j∈ V. This also allows o conside
ha b idges a e ei he open o closed.
The objec i e o his pape is o compu e a se o schedul-
ing decisions u(i,j)
k∈ {0,1}, which a e de ined as ollows:
u(i,j)
k=




1i essel jis scheduled o
pass b idge ia ins an k,
0o he wise.
(1)
Bina y decisions u(i,j)
k, which can also be e e ed o
as manipula ed a iables o con ol inpu s, appea na u ally
in scheduling p oblems o di e en na u e, e.g., mic og id
ope a ion [10], p oduc ion plan scheduling [11] and ma e ial
alloca ion [12]. The objec i e o he dynamic b idge opening
scheduling p oblem is o de e mine u(i,j)
k o sa is y op imal
essel passage imes as much as possible. These decisions
a e hen communica ed o essels, and a e also used o
elabo a e b idge opening schedules, which a e p o ided o
b idges. Vessels and b idges a e assumed o abide by he
scheduling decisions. No e ha decisions ha de ia e om
op imal passage imes may equi e essels o adjus oyage
se ings o comply wi h he solu ion, bu his is ou o he
scope o he pape .
III. PROPOSED APPROACH
The p oposed solu ion o he dynamic b idge opening
scheduling p oblem consis s o wo pa s. A con ol-o ien ed
model o he IWT p oblem in he p esence o mo able
b idges is de i ed i s . Then, he o iginal scheduling p oblem
is ecas as an op imiza ion-based con ol p oblem, and
makes use o he con ol-o ien ed model o de e mine op imal
passage decisions.
A. Con ol-o ien ed model
The wid h occupancy e olu ion o b idge ican be de-
sc ibed using he ollowing disc e e- ime equa ion, ∀i∈ B:
x(i)
k+1 =x(i)
k+X
j∈V(i)
k
(j)u(i,j)
k
| {z }
cu en esou ce booking
−X
j∈V(i)
k−1
(j)u(i,j)
k−1
| {z }
delayed esou ce elease
,(2)
whe e x(i)
k∈R[m] ep esen s he wid h occupancy o b idge
ia ime ins an k,∀i∈ B,∀k∈ K. This occupancy can be
de ined as he amoun o b idge wid h ha is u ilized by
essels o sail h ough a each ime ins an . Mo eo e , (j)
was de ined as he wid h o essel jin Sec ion II.
Equa ion (2) can be iewed as a wid h occupancy bal-
ance, whe eby he cu en occupancy o b idge i, i.e., x(i)
k,
inc eases a he nex ime ins an , i.e, x(i)
k+1, as a esul o
decisions u(i,j)
k= 1,∀j∈ V(i)
k. Howe e , he las assump ion
in Sec ion II s a ed ha essel passage h ough b idges is
done in a single ime s ep. The e o e, inclusion o delayed
con ol ac ions u(i,j)
k−1, which we e de e mined a he p e ious
ime ins an k−1, accoun s o esou ce elease o ese
b idge wid h occupancy.
The se o essels Vwas o iginally de ined as s a ic,
whe eas Vkand Vk−1in Eq. (2) e ince a dynamic na u e. As
essels may en e and lea e he sys em a any ime ins an
k∈ K, he se o essels o be scheduled is ime- a ying
and is he e o e deno ed as Vk, wi h |Vk|=mk. Mo eo e ,
Vkcan be decomposed in o non-o e lapping subse s V(i)
k
such ha Vk=
n
S
i=1
V(i)
k, wi h V(i)
k≜nj:z(i,j)
k= 1o,∀i∈
B,∀k∈ K.
In o de o ack he essel posi ion in a quali a i e manne ,
he e m z(i,j)
kis in oduced o indica e he nex b idge o be
passed by each essel, ∀i∈ B,∀j∈ Vk,∀k∈ K, and is
de ined as ollows:
z(i,j)
k=




1i b idge iis he nex b idge en
ou e o essel ja ins an k,
0o he wise.
(3)
Al hough u(i,j)
kand z(i,j)
kmigh appea somewha simila ,
z(i,j)
k= 1 indica es ha essel jcan pass h ough b idge
ia ime ins an k, while u(i,j)
k= 1 indica es ha essel j
does pass h ough b idge ia ime ins an k,∀i∈ B,∀j∈
Vk,∀k∈ K.
To cap u e b idge- essel passage ope a ions in mo e de ail,
wo addi ional a iables ω(j)
kand s(i)
ka e in oduced. On
he one hand, ω(j)
k∈ {0,1}deno es whe he essel jhas
been scheduled h ough he las b idge be o e eaching he
des ina ion, ∀j∈ Vk, and is de ined as ollows:
ω(j)
k=




1i essel jhas been scheduled
h ough las b idge a ins an k,
0o he wise.
(4)
On he o he hand, s(i)
k∈ {0,1}indica es whe he b idge i
is open o closed a ime ins an k, and is de ined as ollows:
s(i)
k=(1i b idge iis open a ins an k,
0o he wise. (5)
Va iable s(i)
kis in oduced o simpli y he design o ce ain
cons ain s, bu is in ac linked o u(i,j)
k. Al hough his is
discussed la e on in his sec ion, i is con enien o no e he e
ha b idge ishould only be open i and only i a leas one
essel is scheduled h ough b idge ia ha ime ins an .
I should be appa en a his poin ha he scheduling
p oblem should be designed in such way ha logical in-
compa ibili ies a e o bidden. To cla i y his, suppose ha
z(i,j)
k= 1 o a ce ain b idge iand essel ja ime
ins an k. Then, he scheduling p oblem should be endowed
wi h a mechanism ha necessa ily se s u(l,j)
kequal o 0 o
l=i. No e also ha u(i,j)
kmay o may no be se equal
o 1 depending on o he ac o s, e.g., τ(i,j)
e,τ(i,j)
oand o he
ope a ional cons ain s p o ided he eunde .
Logic ules in ol ing he a iables de ined in Eqs. (1),
(3)–(5) can be desc ibed by means o linea equa ions and
(in)equali ies [13]. Se e al p oposi ional logic exp essions
ha cha ac e ize he co ec ope a ion o he sys em a e
iden i ied below o he dynamic b idge opening scheduling
p oblem. Then, he sys ema ic app oach de ailed in [14,
Eqs. (5)–(8)] allows o ans o m a logic exp ession in o
i s equi alen conjunc i e no mal o m. Con e sion o he
esul ing conjunc ion o clauses in o linea (in)equali ies is
hen s aigh o wa d, see [14, Table 1]. Then, he ollowing
logic ules can be s a ed o all b idges i∈ B, essels j∈ Vk
and ime ins an s k∈ K:
•I b idge iis no he nex b idge en ou e o essel
ja ime ins an k, hen essel jcanno be scheduled
h ough b idge ia ime ins an k. This can be o mally
s a ed as: z(i,j)
k= 0→u(i,j)
k= 0. The equi alen
cons ain is
z(i,j)
k−u(i,j)
k≥0,∀i∈ B,∀j∈ Vk,∀k∈ K.(6)
•A ime ins an k, essel jei he has a single b idge
immedia ely en ou e o has al eady been assigned o all
b idges. This can be o mally s a ed as: Pn
i=1 z(i,j)
k⊕
ω(j), whe e ⊕deno es he logical XOR ope a ion. The
equi alen cons ain is
n
X
i=1
z(i,j)
k+ω(j)
k= 1,∀j∈ Vk,∀k∈ K.(7)
•I b idge iis he nex b idge en ou e o essel ja ime
ins an kand essel jis no scheduled h ough b idge i
a ime ins an k, hen b idge iwill be he nex b idge
en ou e o essel ja ime ins an k+ 1. This can be
o mally s a ed as: z(i,j)
k= 1∧u(i,j)
k= 0→
z(i,j)
k+1 = 1. The equi alen cons ain is
−z(i,j)
k+u(i,j)
k+z(i,j)
k+1 ≥0,∀i∈ B,∀j∈ Vk,∀k∈ K.
(8)
•I essel jis scheduled h ough b idge ia ime ins an
kand b idge iis no he las b idge, hen b idge i+1 will
be he nex b idge en ou e a ime ins an k+1. This can
be o mally s a ed as: u(i,j)
k= 1→z(i+1,j)
k+1 = 1.
The equi alen cons ain is
z(i+1,j)
k+1 −u(i,j)
k≥0,∀i∈ B {n},∀j∈ Vk,∀k∈ K.
(9)
•I essel jis scheduled h ough b idge ia ime ins an
kand b idge iis he las b idge, hen essel jhas been
comple ely scheduled a ime ins an k+1. This can be
o mally s a ed as: u(i,j)
k= 1→ω(j)
k+1 = 1. The
equi alen cons ain is
ω(j)
k+1 −u(i,j)
k≥0, i =n, ∀j∈ Vk,∀k∈ K.(10)
•I ea lies passage ime o essel j h ough b idge i
is g ea e han ime ins an k, hen essel jcanno be
scheduled h ough b idge ia ime ins an k. This can be
o mally s a ed as: k≤τ(i,j)
e−1→u(i,j)
k= 0.
The equi alen cons ain is
k≥u(i,j)
kτ(i,j)
e−1+ 1,∀i∈ B,∀j∈ Vk,∀k∈ K.
(11)
•B idge ishould only be open a ime ins an ki
and only i a leas one essel is scheduled h ough
b idge ia ime ins an k. This can be o mally s a ed
as: s(i)
k= 1←→ Pj∈Vku(i,j)
k≥1. The equi alen
cons ain is
s(i)
k≤
mk
X
j=1
u(i,j)
k≤mks(i)
k,∀i∈ B,∀k∈ K,(12)
and mkis he numbe o essels o be scheduled a ime
ins an k.
•I b idge iwas open a ins an k−1and closes a
ins an k, hen b idge imus emain closed du ing
a leas N(i)
down consecu i e ime ins an s. This can be
o mally s a ed as: s(i)
k−1−s(i)
k= 1→s(i)
l= 0.
The equi alen cons ain is
s(i)
k−1−s(i)
k≤1−s(i)
l,∀i∈ B,∀j∈ Vk,∀k∈ K,(13)
and l=k, ..., min k+N(i)
down −1, T , whe e Tis
he scheduling ho izon. In a eceding ho izon con ol
app oach such as he one conside ed in Sec ion III-B,
Tequals he p edic ion ho izon, deno ed as Hp.
Equa ions (11) and (12) a e he esul o implica ions be-
ween a a iable and an inequali y. This equi es o in oduce
a ole ance εand a lowe (uppe ) bound c(C): εcan be se
equal o 1 should he coe icien s and a iables be in ege s
[15, p. 170], and c(C) can be compu ed as he lowe (uppe )
inequali y bound [15, p. 171].
In addi ion o he p e ious cons ain s, he ollowing
physical and ope a ional cons ain s mus also be obse ed:
•Maximum b idge wid h capaci y mus be espec ed:
0≤x(i)
k≤b(i),∀i∈ B,∀k∈ K.(14)
•B idge ican emain open du ing a mos N(i)
up consec-
u i e ime ins an s:
min(k+N(i)
up ,T )
X
l=k
s(i)
l≤N(i)
up ,∀i∈ B,∀k∈ K.(15)
B. Scheduling s a egy: design and implemen a ion
The scheduling s a egy is designed as an op imiza ion-
based con ol p oblem. The e o e, an app op ia e pe o -
mance unc ion is equi ed so ha i s alue can be op imized
while ul illing cons ain s (2), (6)–(15), yielding op imal
scheduling decisions.
Scheduling e o minimiza ion is he ope a ional objec i e
conside ed in his wo k, and can be de ined as he sum o
di e ences be ween op imal passage imes and scheduling
decisions. The quad a ic e o is chosen o be penalized in
his pape , which can be ma hema ically exp essed as
Jk=
n
X
i=1
mk
X
j=1 ku(i,j)
k−τ(i,j)
o2
,∀k∈ K.(16)
Gi en he ac ha u(i,j)
kis dimensionless, i canno be
di ec ly compa ed o τ(i,j)
o, which has disc e e ime uni s.
The e o e, u(i,j)
kis mul iplied by he disc e e ime ins an k.
Then, Jkis minimized when essel jis scheduled h ough
b idge ia ime ins an k=τ(i,j)
o,∀i∈ B,∀j∈ Vk,∀k∈ K.
The model p edic i e scheduling p oblem can hen be
o mula ed as
min
u(i,j)
l|kk+Hp−1
l=k
Ju(i,j)
l|k(17)
subjec o
cons ain s (2),(6)–(15),∀i∈ B,∀j∈ Vk,∀k∈ K,
x(i)
k|k=x(i)
k,∀i∈ B,
z(i,j)
k|k=z(i,j)
k,∀i∈ B,∀j∈ Vk,
ω(j)
k|k=ω(j)
k,∀j∈ Vk,
wi h nu(i,j)
l|kok+Hp−1
l=k
≜nu(i,j)
k|k, u(i,j)
k+1|k,· · · , u(i,j)
k+Hp−1|ko,
whe e k,land k+l|k ep esen he cu en ime ins an , he
ime ins an along he p edic ion ho izon, and he p edic ed
alue o he a iable a ins an k+lusing in o ma ion
a ailable a ins an k, espec i ely. Acco ding o he eceding
ho izon philosophy, only u(i,j)
k|kis applied o he sys em.
P oblem (17) is sol ed again a he nex ime ins an o
u ilize upda ed in o ma ion, hus ans o ming he o iginal
open-loop app oach in o a closed-loop one [16].
Algo i hm 1 ske ches he main implemen a ion de ails
o sol e he dynamic b idge opening scheduling p oblem.
P oblem ini ializa ion is such ha all b idges a e assumed o
Algo i hm 1 Model p edic i e scheduling implemen a ion
Inpu : b(i),N(i)
up ,N(i)
down, (j),τ(i,j)
e,τ(i,j)
o,∀i∈ B,∀j∈ Vk
Ou pu : u(i,j)
k,∀i∈ B,∀j∈ Vk,∀k∈ K
1: Se k= 1 and de ine x(i)
1= 0,z(1,j)
1= 1 and ω(j)
1= 0,
∀i∈ B,∀j∈ V1
2: while mk>0do
3: Design and sol e p oblem (17) conside ing Vk
4: Ex ac u(i,j)
k|kand de e mine x(i,j)
k+1 ,z(i,j)
k+1 ,ω(j)
k+1 and
s(i)
kusing Eqs. (2), (6)–(15)
5: i ω(j)
k+1 = 1 hen
6: Vessel jhas been scheduled h ough las b idge:
dele e om he lis
7: else
8: Vessel jhas no been scheduled h ough las b idge:
keep in he lis
9: end i
10: k←k+ 1
11: Add essels en e ing he sys em a ime ins an k+ 1
o he lis o essels and ini ialize as in S ep 1
12: De ine V(i)
k+1 using he esul o S eps 8 and 11
13: end while
be comple ely a ailable, and all essels mus ini ially pass
h ough he i s b idge. As men ioned be o e, he numbe
o essels o be scheduled a ies o e ime. The e o e, a
new p oblem mus be c ea ed a e e y ime ins an o he
essels p esen in he sys em. This p ocess is epea ed un il
all essels ha e been scheduled h ough all b idges and he e
a e no new essels o be scheduled. Execu ion o Algo i hm 1
concludes when his condi ion is me .
IV. CASE STUDY
The case s udy p esen ed in [9] is used o es he
scheduling app oach p esen ed in Sec ion III. The wa e way
is desc ibed i s , oge he wi h he main ea u es o essels
and b idges. Then, he scheduling solu ion is discussed.
A. Sys em desc ip ion
The Rhine-Alpine co ido connec s majo economic cen-
e s such as B ussels and An we p, he Rands ad egion,
he Rhine-Ruh and Rhine-Necka egions, and Milan and
Genoa. I cons i u es one o he busies Eu opean eigh
ou es, joining he Ro e dam and An we p po s o he
Medi e anean basin. Fu he mo e, i s h oughpu ep esen s
19% o EU’s o al GDP [17].
The Beneden Me wede is a i e s e ch wi hin he
Rhine-Alpine co ido ha uns be ween Do d ech and
Ha dinx eld-Giessendam ( he Ne he lands). A schema ic
ep esen a ion is p o ided in Figu e 1. Da a ega ding oad
and ailway mo able b idges a e p o ided in Table I. Gi en
he small in e -b idge dis ance be ween he i s and second
b idge, hese a e scheduled as a single b idge, and hus he
esul s will be iden ical.
Fi y essels sail om Do d ech o Ha dinx eld-
Giessendam, passing h ough he ou b idges du ing na -
TABLE I
MOVABLE BRIDGES IN THE BENEDEN MERWEDE
B idge (numbe and name) Wid h [m] Maximum up- ime [min] Minimum down- ime [min] App ox. dis ance om p e ious b idge [m]
(1) T a ic b idge Do d ech 44 10 15 –
(2) Railway b idge G o eb ug 44 10 15 50
(3) T a ic b idge Papend ech 30 10 10 4500
(4) Railway b idge Baanhoek 30 15 5 2500
(1)
(2)
(3)
(4)
Fig. 1. Schema ic ep esen a ion o he Beneden Me wede (sou ce:
h ps:// aa wegin o ma ie.nl/)
iga ion. Values o essel wid hs a e aligned wi h he CEMT
class o he wa e way, and ea lies and op imal b idge pas-
sage imes a e gene a ed acco ding o in e -b idge dis ances.
B. Resul s
Passage imes o he i y essels h ough he ou b idges
a e de e mined by applying Algo i hm 1. Resul s a e ob-
ained in Ma lab R2020b using Gu obi Op imiza ion 9.1.2
and YALMIP [18]. A ime s ep size o i e minu es and
a p edic ion ho izon Hp= 1 hou a e selec ed. Al hough
disc e e imes a e deno ed wi h in ege s, hese alues a e
ansla ed in o co esponding i e-minu e ime in e als o
simpli y esul isualiza ion and analysis.
Figu es 2, 3 and 4 depic he scheduling esul s o he
i s and second b idges, hi d, and ou h b idge, espec-
i ely. No e ha in o ma ion p o ided o essels and he
co esponding b idge is shown in he same igu e. On he one
hand, e ical g een ba s ep esen ime slo s du ing which
b idges a e open. On he o he hand, ea lies , op imal and
scheduled essel passage imes a e depic ed as ed, black and
blue ho izon al ba s, espec i ely, and hei wid h equals one
ime s ep, i.e., i e minu es. In he e en ha he scheduled
passage ma ches op imal essel plans, he o e lap is esol ed
by plo ing he scheduled passage ime.
Analysis o he esul s shows ha essels a e scheduled as
close o op imal passage imes as possible while gua an eeing
cons ain ul illmen . All essels a e scheduled a e hei
ea lies passage imes. No essel is scheduled ou side b idge
opening ime ables, and b idges a e only open du ing he
ime s eps essels pass b idges. Maximum up- imes and min-
imum down- imes speci ied in Table I a e espec ed, which
leads o une en ba wid hs in con as o [9]. Maximum
b idge wid h occupancy is espec ed, as shown in Figu e 5.
Fu he mo e, a delay o one sample be ween scheduling
decisions and wid h occupancy o b idges can be no iced
upon inspec ion o Figu es 2–5, in acco dance wi h Eq. (2).
Fig. 2. Fi s and second b idges: τ(i,j)
e( ed), τ(i,j)
o(black), u(i,j)
k(blue)
and opening slo s (g een e ical ba s)
Fig. 3. Thi d b idge: τ(i,j)
e( ed), τ(i,j)
o(black), u(i,j)
k(blue) and opening
slo s (g een e ical ba s)
A quan i a i e esul analysis is ca ied ou on he basis o
he ollowing key pe o mance indica o s (KPIs): pe cen age
o essels scheduled a hei op imal passage ime and ela-
i e b idge wid h occupancy du ing opening (minimum, max-
imum and a e age). The alues a e summa ized in Table II.
On he one hand, i is in e es ing o no e ha sa is ac ion o
op imal passage plans o he la ges b idges, i.e., b idges
1 and 2, a e he lowes . This can be explained—a leas
pa ially—by he ac ha hese wo b idges a e cha ac e ized
by he s ic es maximum up- imes and minimum down-
imes. On he o he hand, ela i e b idge wid h occupancy
shows bo h ha no b idge is o e capaci a ed and ha essel

Fig. 4. Fou h b idge: τ(i,j)
e( ed), τ(i,j)
o(black), u(i,j)
k(blue) and opening
slo s (g een e ical ba s)
Fig. 5. Wid h occupancy o all b idges
passage is ca ied ou in a simila manne o all b idges.
V. CONCLUSIONS AND FUTURE RESEARCH
This pape p esen ed he design o a model p edic i e
scheduling s a egy o coo dina e inland essel na iga ion
and mo able b idge ope a ion o ende wa e bo ne anspo
mo e compe i i e. A con ol-o ien ed model o he p oblem
was o mula ed, paying special a en ion o logic exp essions
ha go e n sys em beha io . A sys ema ic app oach o
con e he exp essions o ma hema ical (in)equali ies was
employed, and he esul ing model was used o c ea e a
TABLE II
VALUES OF KEY PERFORMANCE INDICATORS (KPIS)
KPI Value (in pe cen age)
B idges 1 and 2 B idge 3 B idge 4
Pe cen age o essels scheduled
a hei op imal passage ime 34 48 72
Rela i e b idge
wid h occupancy
Minimum 15 16.83 16.83
Maximum 96.14 98 98.83
A e age 55.41 49.25 49.16
p edic i e scheduling s a egy ha de e mined essel passage
imes and b idge ope a ion ime ables ensu ing coo dina ion.
Se e al esea ch a enues can be explo ed on he basis
o he esul s p esen ed in his pape . On he one hand,
essel oyage plans a e cha ac e ized by a ce ain deg ee
o unce ain y, which may be agg a a ed by he p esence
o essels ha do no pe o m V2I communica ion, e.g.,
ec ea ional boa s. Robus and s ochas ic con ol app oaches
will be conside ed o mi iga e he unce ain y. On he o he
hand, ope a ional objec i es om he s andpoin o b idges
will be included in he cos unc ion, and he use o Pa e o
op imiza ion will be explo ed o de e mine sa is ac o y ade-
o solu ions.
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