A model predictive scheduling strategy for coordinated inland vessel navigation and bridge operation

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. 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