CONTAINER BERTH SCHEDULING POLICY WITH VARIABLE COST FUNCTION

Contner berth schedulng polcy wth vrble cost functon CONTAINER BERTH SCHEDULING POLICY WITH VARIABLE COST FUNCTION Gols M.M. (correspondng uthor) Assstnt Professor, Deprtment of Cvl Engneerng, Unversty of Memphs, Memphs, 3815 Centrl Ave Memphs, TN 38152, USA, Phone: 901-678-3048, Fx: 901-678-3026, Eml: mhlsgols@yhoo.com Hrlmbdes H.E. Professor of Mrtme Economcs & Logstcs, Center for Mrtme Economcs & Logstcs (MEL), Ersmus School of Economcs (Econometrc Insttute), Ersmus Unversty Rotterdm, P.O. Box 1738, 3000 DR Rotterdm, The Netherlnds, Tel: +31-10-408-1484, Fx: +31-10-408-9093, Eml: hrlmbdes@ese.eur.nl Abstrct: Ths pper presents new mthemtcl formulton for the berth schedulng problem. The obectve s to smultneously mnmze the totl cost of vessels lte deprtures nd wtng tme, nd mxmze the benefts from vessels erly deprtures. It s ssumed tht dfferent vessels, belongng to the sme or dfferent lner shppng compnes, hve dfferent contrctul greements, nd thus dfferent cost functons. We dscuss the pplcblty nd dvntges of nonlner cost functons, vs à vs the lner ones commonly used n the lterture. A genetc lgorthm bsed heurstc s proposed to solve the resultng problem nd number of computtonl exmples re presented to crtclly ssess the proposed berth schedulng polcy nd evlute the effect of the ssumed cost functon on the spred nd dstrbuton of the totl cost mong ll vessels. Keywords: Mrne Contner Termnls, Berth Schedulng, Cost Functon, Metheurstc Optmzton INTRODUCTION The berth schedulng problem (BSP) dels wth the ssgnment of vessels to berth spce n contner termnls (Theofns et l., 2009; Mesel, 2009; Berwrth nd Mesel, 2009). Among the vrous models found n the lterture, three ssumptons re usully observed: ) dscrete vs. contnuous berthng spce; b) sttc vs. dynmc vessel rrvl, nd c) sttc vs. dynmc vessel hndlng tme. In the dscrete problem, the quy s vewed s fnte set of 1

Contner berth schedulng polcy wth vrble cost functon berths (see for exmple Hnsen et l., 2007; Im et l., 1997; Im et l., 2001; Im et l., 2003; Monco et l., 2007). In the contnuous cse, vessels cn berth nywhere long the quy (see for exmple Prk nd Km, 2003; Km nd Moon, 2003; Gun nd Cheung, 2004; Im et l., 2005; Moorthy nd Teo, 2006; Mesel nd Berwrth, 2009). The morty of the publshed reserch consders the dscrete cse (Theofns et l., 2009; Mesel, 2009; Berwrth nd Mesel, 2009). In the sttc rrvl problem, ll vessels to be served re lredy n the port t the tme schedulng begns. In the dynmc rrvl problem, not ll vessels to be scheduled for berthng hve rrved, lthough rrvl tmes re known n dvnce. The morty of the publshed work on berth schedulng consders the ltter cse. Fnlly, n the sttc hndlng tme problem, vessel hndlng tme s consdered s known nput, wheres n the dynmc formulton, vessel hndlng tme s vrble, usully ssumed functon of the quy crnes tht wll operte on the vessel nd the dstnce between the berthng poston nd locton n the yrd. (Berwrth nd Mesel, 2009). Techncl restrctons, such s berthng drft, nter-vessel nd end-berth clernce dstnce re further ssumptons tht hve been dopted n some studes, n n ttempt to brng problem formulton closer to rel world condtons. The ntroducton of techncl restrctons to exstng berth schedulng models s rther strghtforwrd nd s therefore not ttempted here. The pper presents new formulton for the dscrete spce - dynmc rrvl tme berth schedulng problem (DDBSP). The obectve s to smultneously mnmze the totl cost from vessels lte deprtures nd wtng tme, nd mxmze the benefts from vessels erly deprtures. Vessel deprture tmes re often determned through contrctul greements between the termnl opertor nd the crrer, n relton to the shp s rrvl t the port nd the totl number of contners to be (un)loded. To dte, models found n the lterture hve ssumed tht penltes (or premums n cse of erly deprtures) for lte deprture contnue to ncrese ndefntely wth tme (usully on n hourly bss). Ths s not lwys the cse nd n prctce two ddtonl cses cn be observed: ) f vessel s trdy, the termnl opertor pys fxed penlty rrespectve of the length of the dely, nd b) the termnl opertor pys fxed penlty up to pont n tme nd he swtches to n hourly rte fter ths. These three cses my be observed smultneously, s dfferent vessels my hve dfferent contrctul greements. All three cses re therefore ncluded n ths pper, ssumng tht dfferent vessels, belongng to the sme or dfferent crrers, hve dfferent contrctul greements, nd thus dfferent cost functons 1. Ech vessel s ncluded n the obectve functon under one of the three polces (.e. vessel cnnot be subect to two dfferent polces). The effect of the three polces s evluted nd prelmnry results re presented. At the sme tme, we lso evlute the effect of nonlner cost functons -s opposed to lner ones used to dte- on the dstrbuton of costs mong vessels. A genetc lgorthm (GA) bsed heurstc s employed to solve the resultng problem. An ndctve number of computtonl exmples re presented to dscuss the dfferent polces nd ther effect on the cost functon. As noted n Hrlmbdes (2002), dfferences exst n berth schedulng polces between mult-user nd dedcted termnls (see lso Avers et l., 2005). The berthng polces proposed here re better suted for publc, mult-user, termnls but they could eqully pply to dedcted termnls s prt of mult-obectve or herrchcl formulton (Gols et l. 2009c; Gols et l., 2009d). 1 The term cost functon n ths pper refers only to the trdy penltes nd erly premums of deprtures nd should not be confused wth vessel s totl cost functon or tht of the termnl opertor (Hrlmbdes, 2004). 2

Contner berth schedulng polcy wth vrble cost functon The rest of ths pper s structured s follows. The next secton presents bref descrpton of models nd obectves publshed to dte on the DDBSP, followed by secton wth the problem descrpton nd ts mthemtcl formulton. The fourth secton presents the proposed resoluton lgorthm used to solve the formulted problem. The ffth secton presents results from number of computtonl exmples nd the lst secton concludes the pper nd suggests future reserch drectons. LITERATURE REVIEW Im et l. (1997) were the frst to formulte the dscrete spce - sttc rrvl tme berth schedulng problem s n unrelted mchne schedulng problem. They ssert tht, n ports wth hgh throughput rtes, optml vessel-to-berth nd vessel-to-tme-slot ssgnments should be found tht go beyond the frst come frst served (FCFS) rule. They thus develop heurstc soluton pproch to solve the resultng problem. Im et l. (2001) subsequently extended ther 1997 work to ddress the DDBSP. A Lgrngn relxton heurstc ws proposed nd computtonl experments showed tht the heurstc performs well n prctce. Nshmur et l. (2001) ddress the sme problem usng Genetc Algorthms s resoluton pproch. It should be noted tht the resoluton pproch used by Nshmur (.e. GA bsed heurstcs) hs been ppled extensvely by dfferent reserch groups n the followng yers (Bole et l., 2009). Im et l. (2003) modfed nd extended the formulton of Im et l. (2001) by ncludng servce prorty constrnts. The problem ws reformulted nto qudrtc ssgnment problem, usng Lgrngn relxton method, nd GA heurstc ws proposed s the soluton pproch. Cordeu et l. (2005) consder the DDBSP, presentng two formultons (one bsed on the Mult Depot Vehcle Routng Problem wth Tme Wndows nd one bsed on the model by Im et l. (2001)). A Tbu Serch heurstc ws proposed s the soluton pproch. Brno et l. (2005) outlne the ntegrton of flexble smultor, representng the se-sde opertons of contner termnl, wth lner progrmmng model for mprovng berth ssgnment nd yrd stckng polces. Lokuge nd Alhkoon (2007) present unque pproch, deprtng from ll prevous work. They use Artfcl Intellgence (AI), nd more specfclly the Belefs, Desres nd Intenton (BDI) gent rchtecture for vessel berthng pplcton system. Results showed reduced verge wtng tme of vessels, whle severl other mesures of port productvty were lso presented. Smlrly to erler reserch, optmlty of the fnl schedule ws not gurnteed. Moorthy nd Teo (2006) present novel pproch for the DDBSP, whch for the frst tme ncorported the stochstc nture of vessel rrvls, wth very promsng lbet not optml results. Hn et l. (2006) present non-lner modelng formulton nd propose soluton pproches bsed on GAs nd combnton of GAs nd smulted nnelng. Nether pproch gurnteed optmlty. Monco nd Smr (2007) formulte the DDBSP s dynmc schedulng problem of unrelted mchnes. They develop new non-stndrd multpler dustment Lgrngn heurstc lgorthm. Im et l. (2007b) propose bobectve formulton to mnmze shp delys nd totl servce tme. They propose GA bsed lgorthm, s well s subgrdent optmzton procedure. The uthors use lner cost functon. Lee nd Chen (2009) propose formulton for cse where berths re ssgned to blocks of storge; cse between the dscrete nd contnuous berthng spce BSP, bsed on the FCFS prncple. A neghborhood serch bsed pproch ws proposed. The novelty of ths reserch ws tht t could hndle lrge scle problems wthn smll CPU tme, lthough optmlty, or comprson of the fnl schedule wth lower bounds, ws not provded. 3

Contner berth schedulng polcy wth vrble cost functon Im et l. (2007) ddress the berth llocton problem t mult-user contner termnl wth ndented berths for fst hndlng. A GA heurstc ws gn ppled s the resoluton pproch. Im et l. (2008) ddress vrton of the DDBSP t mult-user termnls, where vessels exceedng n expected wtng tme lmt re ssgned to n dcent termnl. A GA bsed heurstc ws proposed for the resoluton of the problem. Hnsen et l. (2008) studed the DDBSP, consderng the mnmzton of totl costs of wtng nd hndlng, s well s erlness or trdness of completon, for ll vessels. A Vrble Neghborhood Serch heurstc ws proposed nd compred wth Mult-Strt; GA; nd Memetc Serch Algorthm. Smlrly to Im et l. (2007b), the premum nd cost functon ws lner. Gols et l. (2009d) formulted the berth schedulng problem s b-level unrelted mchne schedulng problem wth vrble vessel relese dtes to ccommodte n envronmentlly frendly BSP, whle preservng the ntegrty of the ocen crrers schedule. An evolutonry lgorthm bsed heurstc ws proposed s resoluton pproch. Gols et l. (2009b) studed the DDBSP wth the obectve of smultneously mnmzng delyed deprtures nd mxmzng erly nd on-tme deprtures/berthng of vessels wthn tme wndow. An dptve tme wndow prttonng heurstc ws proposed s the resoluton pproch tht dd not gurntee optmlty of the fnl schedule. Cheong nd Tn (2008) formulte smlr BSP s Im et l. (2007b) but s non-lner b-obectve problem wth lner dely cost functons. Cheong et l. (2009) formulte the DDBSP s non-lner three-obectve problem mnmzng totl mkespn, totl wtng tme, nd devtons from predetermned prorty. Gols et l. (2009c) formulte the DDBSP for the frst tme s mult-obectve problem wth n+1 obectve functons (where n s less thn or equl to the number of vessels) to provde customer-bsed dfferentted servces bsed on vessel servce tme. Fnlly, Gols et l. (2009) propose lmd-optmzton bsed heurstc for the resoluton of the DDBSP wth promsng results tht gurntee locl optmlty n pre-specfed neghborhood. MODEL FORMULATION From the bove lterture revew t cn be concluded tht most of the studes delng wth the DDBSP hve focused on totl servce nd wtng tme (totl completon tme), s well s on the costs/premums from delyed/erly deprtures. The ltter obectves were ntroduced to tke nto ccount contrctul greements on the scheduled strt or fnsh tme of shp s crgo hndlng opertons. Such rrngements cn vry from berthng upon rrvl, to gurnteed servce tme wndow, nd/or gurnteed servce productvty. Erlness or delys n crgo hndlng opertons mply benefts or costs to both the termnl opertor nd the ocen crrer (Hrlmbdes, 2002b). If crgo hndlng s completed fter the greed tme (deprture dedlne), the opertor my py penlty to the crrer, whle, n the opposte (erly deprture), the crrer my py premum fee to the termnl opertor. Erly deprture cn help crrer mnge tme to next port of cll, smply by offerng buffer to compenste for tme lost n other ports (Notteboom, 2006). Premums pd by the crrer to the termnl opertor, on the other hnd, cn be offset by reducng voyge costs through lower voyge speed nd therefore fuel consumpton. Slow-stemng s currently becomng n ncresngly ttrctve polcy mong crrers, n ther efforts to bsorb superfluous excess cpcty nd thus mntn rtes t n cceptble level. Industry observers lso note tht slow-stemng my even be sttutorly determned n the future for envronmentl resons. As lredy mentoned bove, reserch so fr hs ssumed hourly bsed erly deprture premums or trdy deprture penltes. We ssume here tht dfferent vessels, belongng to 4

Contner berth schedulng polcy wth vrble cost functon the sme or dfferent lner shppng compnes, hve dfferent contrctul greements nd thus dfferent cost functons. It s lso ssumed tht only one of the followng three greements cn pply to shp (fgure 1), where the termnl opertor: pys (receves) fxed penlty (premum) rrespectve of the length of the dely (erlness) [cost polcy 1]. pys (receves) lner hourly penlty (premum) ccordng to the length of the dely (erlness) [cost polcy 2]. pys (receves) constnt penlty (premum) up to pont n tme beyond whch he pys (receves) n hourly penlty (premum) [cost polcy 3]. Fgure 1 - Illustrton of dfferent cost/premum greements In order to formulte the berth schedulng model (BSM) we need to defne the followng: 5

Contner berth schedulng polcy wth vrble cost functon Sets I : set of berths J : set of vessels J J 1 : set of vessels under fxed premum/cost greement for erly/lte deprture J J 2 : set of vessels under hourly premum/cost greement for erly/lte deprture J J 3 : set of vessels under both fxed nd hourly premum/cost greement for erly/lte deprture Decson Vrbles x y b f l { 0,1}, I, J =1 f vessel s served t berth nd 0 otherwse { 0,1},, b J, b =1 f vessel b s served t the sme berth s vessel α s ts mmedte successor nd 0 otherwse { 0,1 }, J =1 f vessel s served s the frst vessel { 0,1 }, J =1 f vessel s served s the lst vessel Auxlry Vrbles HED R J 1 J : totl hours the vessel deprts before the requested dedlne, 2 HLD R J 1 J : totl hours the vessel deprts fter the requested dedlne, 2 ed { 0,1 } J =1 f vessel deprts before the requested dedlne nd 0 otherwse ld { 0,1 } J =1 f vessel deprts fter the requested dedlne nd 0 otherwse t, J : strt tme of servce for vessel lph R J : postve number Prmeters ED R J : erly requested deprture dedlne tme of vessel (pplcble only n the thrd cost polcy) 6

Contner berth schedulng polcy wth vrble cost functon LD R J : lte requested deprture dedlne tme of vessel (pplcble only n the thrd cost polcy) RDT R J : requested deprture dedlne tme of vessel hp R J : hourly premum f vessel deprts before the requested dedlne ( 1 hp 0, J ) hc R J : hourly penlty f vessel deprts fter the requested dedlne ( 1 hc 0, J ) fp R J : fxed premum f vessel deprts before the requested dedlne ( fp 0, J2) fc R J : fxed penlty f vessel deprts fter the requested dedlne ( fc 0, J2) wtc R J : hourly wtng tme cost for vessel S A R I : tme berth becomes vlble for the frst tme n the plnnng horzon R J : rrvl tme of vessel α: degree of non-lner cost functon The berth schedulng model (BSM) proposed n ths pper cn thus be formulted s follows: mn wtc ( t A ) fc ld fp ed J hc HLD hp HED hc [ HLD RDT LD ] hp [ HED ED RDT ] J 3 Subect To: J1, J3 Decson vrble constrnts x 1, J (2) J 2 (1) f b y b b 1, b J (3) 7

l l f y b y b Contner berth schedulng polcy wth vrble cost functon 1, J (4) f 3 x x, I,, b J, b (5) b b l 3 x x, I,, b J, b (6) b ed b b x xb 1 yb, I2,, b J, b (7) 1 2 ld 1, J (8) Vessel strt tme estmton t t t b A, J (9) S x, I, J (10) t c x M ( 1 y ),, b J, b (11) b Erly deprture estmton ED t c x M ( 1 ed ), J (12) ED ( 1 ed ) t c x, J (13) HED ED ed t c x lph, J (14) lph M (1 ed ), (15) lph t c x, (16) Lte deprture estmton RTD t c x M ( 1 ld ), J (17) RTD ( 1 ld ) ld M t c x, J (18) HLD t c x RTD, J (19) 8

Contner berth schedulng polcy wth vrble cost functon Fgure 2 - Estmton of hours of erly/lte deprture before/fter erlest/ltest request The obectve functon (1) mnmzes the totl cost of vessels wtng tme nd lte deprtures, nd mxmzes the totl premums from erly deprtures. The frst component of the obectve functon mnmzes the totl cost of vessels wtng tme. The second component mnmzes the totl trdy cost nd mxmzes the erly deprture premum for vessels under the fxed cost/premum polcy. The thrd nd fourth components of the obectve functon do the sme for vessels under the hourly- nd fxed nd hourly cost/premum polces respectvely. It hs often been rgued (Gupt nd Sen, 1983; Su nd Chng, 1998; Schller, 2002) tht lner cost functons encourge schedules n whch only few shps contrbute to the morty of the cost, wth no regrd to how the overll cost s dstrbuted. To tke ths chrcterstc nto ccount, non-lner obectve functon (eq. 1) s, dopted here, wth ech component rsed to the power of α 2. The ssumpton tht non-lner functon would dstrbute the cost more evenly s evluted n the next secton through number of numercl exmples for dfferent degrees of the non-lner functon (ncludng the lner cse). Constrnt set (2) ensures tht ech vessel s served once, whle constrnt set (3) ensures tht ech vessel s ether served frst or s preceded by nother vessel. In smlr mnner, constrnt set (4) ensures tht ech vessel s ether served lst or before nother vessel. Constrnt sets (5) nd (6) ensure tht only one vessel cn be served frst nd lst t ech berth. Constrnt set (7) ensures tht vessel cn be served fter nother vessel, only f both re served t the sme berth. Constrnt set (8) ensures tht vessel wll ether deprt erly or lte. Constrnt sets (9) nd (10) ensure tht vessel servce strt tme s greter thn vessel rrvl, or the tme tht the berth where the vessel wll be served becomes vlble for the frst tme n the plnnng horzon. Constrnt set (11) estmtes the servce strt tme of ech vessel. Constrnt sets (12) through (16) estmte the totl hours of erly deprture. If vessel does not deprt erly from berth (.e. ED t c x ) then ed wll tke the vlue of zero so tht equton (12) s fesble (.e. ED M ). In tht cse, equtons (15) nd (16) set lph to ts upper bound (.e. lph t cx ), equton (13) s fesble s n 2 In the obectve functon, the constnt penltes/premums (.e. second term) re rsed to the sme power s the rest of the components to vod dmnshng nfluence (of ths term) to the obectve functon (nd thus vessel-toberth ssgnment) s the degree (α) ncreses. 9

nequlty (.e. Contner berth schedulng polcy wth vrble cost functon ED t c x ), nd equton (14) sets HED to ts lower bound (.e. zero). If the vessel does deprt erly (.e. one so tht equton (13) s fesble (.e. fesble s well (.e. ED t c x ED t c x ), then ed wll tke the vlue of 0 t c x ). In tht cse, equton (12) s ). The cse where vessel, served t berth I, deprts erly s smlr nd thus omtted. Constrnt sets (17) through (19) estmte the totl hours of lte deprture. If vessel does not deprt lte from berth (.e. RTD t c x ) due to constrnt set (8) ld s equl to zero. In ths cse constrnt set (17) s fesble (.e. RTD M nd constrnt set (19) sets HLD equl to the lower bound (.e. zero). The cse where vessel, served t berth I, deprts lte s smlr nd thus omtted. RESOLUTION ALGORITHM The BSM s non-lner mxed nteger problem (MIP) for α>1 nd lner MIP for α=1. In both cses, no exct resoluton lgorthm exsts to dte tht cn solve such problems n polynoml tme (especlly n rel lfe nstnces whch on verge hve mnmum of 10 to 15 vessels nd 3 to 5 berths). To tckle ths, the GA bsed heurstc, proposed by Gols (2007), s employed here s the resoluton pproch. The proposed heurstc conssts of four prts: ) the chromosoml representton; b) the chromosoml mutton; c) the ftness evluton; nd d) the selecton process. The GA uses n nteger chromosoml representton, n order to explot n full the chrcterstcs of the problem. An llustrton of the chromosome structure s gven n fgure 3 for smll nstnce of the problem wth 6 vessels nd 2 berths. As seen n fgure 3, the chromosome hs twelve cells. The frst 6 cells represent the 6 possble servce orders t berth1 nd the lst 6 cells the 6 possble servce orders t berth 2. In the ssgnment llustrted n fgure 2, vessels 2, 4, nd 5 re served t berth 1 s the frst, second nd thrd vessel respectvely, whle vessels 1, 3, nd 6 re served t berth 2 s the frst, second, nd thrd vessel respectvely. Fgure 3 - Illustrton of chromosome representton Four dfferent types of mutton re ppled, s prt of the genetc opertons, to ll the chromosomes t ech generton: nsert, swp, nverson, nd scrmble muttons. Ech of the four types of mutton, llustrted n fgure 3, for the smll exmple shown n fgure 4, hs ts own chrcterstcs n terms of preservng the order nd dcency nformton (Eben nd Smth, 2003). 10

Contner berth schedulng polcy wth vrble cost functon Fgure 4 - Schemtc llustrtons of the mutton opertons Snce the cse t hnd s mnmzton problem, the smller the vlues of ech obectve functon, the hgher the ftness vlue. As dscussed n Goldberg (1989), n mnmzton problems t s desrble to defne the ftness functon of chromosome s: pt pt f ( x) Fmx z ( x), where F mx s the mxmum vlue of the obectve functon z t, nd f t pt pt s the vlue of the ftness functon of obectve functon t terton t. However, ths vlue s not known n dvnce; thus the lrgest vlue z pt for ech obectve functon t ech terton pt s chosen s the vlue of F mx. To fnd qulty solutons for ech obectve functon nd, t the sme tme, retn vrety of dfferent solutons, the Roulette Wheel Selecton lgorthm (Goldberg, 1989) ws ppled to select the chldren populton of ech generton. COMPUTATIONAL EXAMPLES Fve problem nstnces re developed, where vessels wth vrous hndlng volumes re served t mult-user contner termnl wth fve berths, plnnng horzon of one week, nd vessel nter-rrvl tme of 3 hours. The rnge of the remnng prmeters consdered here s chosen ccordng to Gols et l. (2009c) nd Hnsen et l., (2008) nd they re reported for purposes of consstency. Avlblty of berths for the frst tme n the begnnng of the plnnng horzon (.e. prmeter S ) s clculted usng unform probblty dstrbuton wth mnmum of zero nd mxmum of 10 hours (Hnsen et l., 2008). Delyed nd erly deprture; hourly nd constnt penltes; nd hourly wtng costs re clculted rndomly, not bsed on ctul crgo crred, s vessels crryng less thn cpcty crgo mght belong to crrer wth hgher prortes. Ltest nd erlest deprture requests re generted rndomly bsed on the formul by Hnsen et l. (2008). Vessel hndlng volumes re generted rndomly bsed on unform dstrbuton pttern (Gols et l., 2009c) between 5 nd 30 hours t the preferred berth (.e. berth wth the mnmum hndlng tme over ll berths). The mnmum vessel hndlng tme t berth (excludng the preferred berth) s generted n relton to the berth wth the mnmum hndlng tme, by ncresng hndlng tme proportonlly to the dstnce from the preferred berth (Im et l., 2001). The ncrese s bsed on unform probblty dstrbuton. Preferred berths for the vessels re chosen 11

Contner berth schedulng polcy wth vrble cost functon rndomly. Experments were performed on Dul Core ASUS CM5570 computer wth 6GB memory, usng Mtlb R2008b 3. The ntl populton for the GA bsed heurstc ws obtned by usng frst come frst served (FCFS) rule, t the frst vlble berth (smlr to Hnsen et l., 2008). The populton sze ws set to 50 chromosomes nd the heurstc stops f no mprovement s observed for 100 tertons (.e. new or mproved schedules found). Berth Schedulng Polcy Evluton Usng the dtset descrbed n the prevous subsecton, we performed two types of experments. The frst focuses on the effect of the degree of the non-lner cost functon, nd the second on the effect of the cost polcy on the dstrbuton of totl cost mong vessels. To perform these experments, for ech of the fve dtsets presented bove, 16 dfferent berth schedules were consdered. Ech berth schedule ws obtned by usng dfferent combnton of cost polcy nd degree of cost functon, keepng the remnng dt of ech dtset unchnged (.e. hndlng tme; rrvl tme; hourly nd constnt penltes; etc.). Four dfferent degrees were used for the cost functon, wth vlues of α=1, 2, 3, nd 4. For the type of cost polcy, we ssume the followng four dfferent cses: ) Cse 1: ll vessels fll under cost polcy 1; b) Cse 2: ll vessels fll under cost polcy 2; c) Cse 3: ll vessels fll under cost polcy 3; d) Cse 4: ech vessel flls rndomly under one of the three cost polces, bsed on unform probblty dstrbuton. It should be noted tht lthough schedules were obtned usng the non-lner cost functon (.e. α=1, 2, 3, nd 4), the cost vlues used n the evluton were bsed on the lner cost functon of ech schedule (.e. α=1). To obtn mesure of unformty of the cost dstrbuton mong ll vessels, unform dstrbuton ws ftted, usng s observtons the cost of ech vessel from ech schedule. Fgure 4 shows the rnge of the estmted prmeters of the ftted unform dstrbuton (.e. mnmum nd mxmum vlue) for the 16 dfferent schedules for ech dtset. Fgure 5 shows the vrnce of the ftted unform dstrbutons for the 16 dfferent schedules of ech dtset. The x-xs represents the degree of the obectve functon, wth totl cost mesured on the y-xs. Ech grph plots four lnes, one for ech cse prevously presented. 3 www.mthworks.com 12

Contner berth schedulng polcy wth vrble cost functon Fgure 4 - Dfferences n upper nd lower bound of unform dstrbuton We observe tht, n generl, s the degree of the non-lner functon ncreses (.e. movng rght on the x-xs) the spred nd vrnce of totl cost mong vessels decreses n ll cses. Ths becomes more notceble, s the totl cost dfference between the smllest nd lrgest vlue of the ftted dstrbuton ncreses (e.g. fgures 4 nd 5, dtsets 1 nd 2, cost polces 2 nd 3), supportng our ntl ssumpton tht non-lner cost functons spred the cost mong ll vessels more evenly. On the other hnd, one notes tht obectve functon vlues (shown n fgure 6 for the 16 schedules per dtset), ncrese, n generl, wth the degree of the cost functon. In fgure 6, the y-xs shows the vlue of ech obectve functon s percentge of ts mxmum vlue mong the four dfferent degrees, for the sme cse. For exmple, for dtset 1, the obectve functon vlues for schedules wth cost functons of degrees α=1, α=2, nd α=3 (for the second cse) were 21%, 7%, nd 2% lower thn the obectve functon vlue of the berth schedule wth α=1 (for the sme cse). 13

Contner berth schedulng polcy wth vrble cost functon Fgure 5 - Chnges n the vrnce CONCLUSIONS We hve presented mthemtcl formulton for the berth schedulng problem where the totl cost from vessels lte deprture nd wtng tme s mnmzed, nd totl benefts from erly deprtures mxmzed. Dfferent cost functons for ech vessel were used, to represent dfferent contrctul greements. To the best of our knowledge ths formulton nd cost polces re ttempted for the frst tme n the publshed lterture. We hve lso dscussed the pplcblty nd effectveness of nonlner cost functons, nd evluted the ssumpton tht non-lner functon would dstrbute costs more evenly mong vessels. A GA bsed heurstc ws used to solve the resultng problem, nd number of computtonl exmples showed tht hgher order degrees of the cost functon provde smller devtons of the spred nd vrnce of totl cost mong vessels but result n hgher cumultve cost. 14

Contner berth schedulng polcy wth vrble cost functon Future reserch could focus on: ) performng lrger number of computtonl exmples wth ncresed sze (.e. up to ten berths nd seven hours of vessel nter-rrvl tmes), b) nvestgte formulton tht explctly models the vrblty nd dsperson of costs mong vessels nd the totl cost (.e. b-obectve or b-level formulton), nd c) ntroduce tme wndow deprture request whereby f the vessel deprts wthn tme wndow, fter ts rrvl, no penltes or premums re pplcble. Fgure 6 - Chnges n the obectve functon vlue 15

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