Multi-objective simultaneous stowage and load planning for a containership with container rehandle in yard stacks

Transcription

1 Mult-objectve smultaneous stowage and load plannng for a contanershp wth contaner rehandle n yard stacks Ako Ima Faculty of Martme Scences Kobe Unversty Fukae, Hgashnada, Kobe Japan & World Martme Unversty, PO Box 500, S Malmo, Sweden Kazuya Sasak, Etsuko shmura Faculty of Martme Scences Kobe Unversty Fukae, Hgashnada, Kobe Japan and Stratos Papadmtrou Department of Martme Studes Unversty of Praeus 40, Karaol & Dmtrou Str., GR Praeus, Greece

2 Abstract The effcency of a martme contaner termnal prmarly depends on the smooth and orderly process of handlng contaners, especally durng the shp s loadng process. The stowage and assocated loadng plans are manly determned by two crtera: shp stablty and the mnmum number of contaner rehandles requred. The latter s based on the fact that most contaner shps have a cellular structure and that export contaners are pled up n a yard. These two basc crtera are often n conflct. Ths paper s concerned wth the shp s contaner stowage and loadng plans that satsfy these two crtera. The GM, lst and trm are taken nto account for the stablty measurements. The problem s formulated as a mult-objectve nteger programmng. In order to obtan a set of nonnferor solutons of the problem, the weghtng method s employed. A wde varety of numercal experments demonstrated that solutons by ths formulaton are useful and applcable n practce. Keywords: Logstcs; Multple objectve programmng; Genetc algorthms; Heurstcs; Contanershp handlng 1

3 1. Introducton The overwhelmng majorty of general cargo s nowadays contanerzed. As the contanerzed transportaton system s captal-ntensve, the fast shp turnaround at a contaner termnal s essental for the economc performance of lner shppng companes. The turnaround tme of a shp ncludes the tme for berthng, unloadng, loadng and departure, and therefore t can be stated that contaner loadng and unloadng are crtcal elements of the effcency of ths transport chan. Shp operators as well as port managers are keen n determnng the optmal vessel stowage and assocated loadng plans, whch mnmze shp s dwell tme n port wth acceptable shp stablty. In a cellular contanershp, f specfc contaners (referred to as target contaners) must be stowed at vertcally mddle locatons n a shp s hold for stablty reasons, they have to be loaded n a loadng sequence after the contaners that are to be stowed below them and before the contaners that are to be stowed above them. Concurrently, another restrcton emerges durng the pckng of contaners from a yard to be loaded onto the shp, snce contaners are pled up to form block formatons on the yard for storage purposes. If the target contaners are stacked on the yard below others, whch are to be pcked up later, then the loadng task requres the so-called loadng-related rehandle n order to remove and reposton the others. Ths s very lkely to occur, as detaled nformaton about the order of the loadng sequence s not avalable when contaners begn to arrve at the termnal from the nterland. Furthermore, even when the loadng nformaton s avalable, the deal layout of export contaners n the storage area of the yard s almost mpossble to be acheved due to the random arrval of contaners. A way to avod rehandle durng a loadng operaton, would be contaner shufflng n advance of loadng n order to group the contaners by destnaton and weght. However, ths necesstates addtonal workload for the handlng equpment of an enormous scale, because the whole set of contaners to be loaded must be arranged, probably n other stacks, n order to orderly remove them from the stacks for the loadng operaton, wthout any unproductve rehandles. otce that ths task could be performed only when the handlng equpment s dle. Otherwse t would conflct wth the ongong tasks of loadng other shps. In addton, smooth shufflng may requre a buffer stackng area, where contaners 2

4 to be loaded are moved orderly from the storage area, whch accommodates ncomng contaners from shppers. However, such a buffer area seems hardly practcal or realstc for land scarce contaner termnals. There s another type of rehandle, unloadng-related rehandle, whch refers to movng contaners onboard that are not destned for beng dscharged at a partcular port, to reach others that are to be unloaded at that port. Ths s lkely to occur when contaners destned for a specfc port are spread out over several shp holds, beng assocated wth the shp routng and dfferent types of contaners on board. The unloadng-related rehandle may occur when lmted shp capacty or strct stablty condton n a complex tnerary of callng port requres contaners wth dfferent destnatons to be stowed mxed n a partcular vertcal column. Ths paper s concerned wth stowage and assocated load plannng of a contanershp whle satsfyng the shp s stablty such as GM (the dstance between the center of gravty and the metacenter poston), lst and trm, whle mnmzng the number of contaner rehandles. We focus only on the loadng-related rehandle; therefore ths problem may be restrctve n practcal use. However, t s applcable n some cases for the followng reasons: Our survey found that for several deep-sea shppng lnes of Japan, each hold of a shp or a vertcal column n a hold normally stows contaners that are all destned for a partcular port because the shp has few ports of call, resultng n fewer occurrences of the unloadng-related rehandle. Ths s very lkely especally when the shp s over-capactated. In addton, even though a voyage calls at a number of ports, only a lmted number of rehandles lkely occur f the voyage tnerary forms the pendulum-type routng (a number of deep-sea routes are the pendulum) such as an tnerary of callng port: A-B-C-D-E-D-C-B-A. In deep-sea routes connectng two regons such as Asa-Europe and Asa-orth Amerca, lttle ntra-regonal traffc s observed. For nstance, assume that ports A, B and C are stuated n Japan and Korea whle D and E are located n US. Deep-sea vessels move very few contaners among A, B and C, and no traffc between D and E because of cabotage. Wth such a trade pattern, contaners for the farthest destnaton port of US n the callng sequence should be loaded, at each orgn port n Asa, under those for the nearest destnaton port of US; and then, n the return voyage the same stowage arrangement can be appled because each port s called 3

5 twce. Ths scheme results n qute few or even no unloadng-related rehandles. The shp s loadng sequence assocated wth the stowage problem may affect a shp s handlng tme, snce an neffcent loadng sequence forces the handlng equpment, especally the transtaner, to make redundant travels (or moves). Consequently, the stowage problem mght nclude the loadng sequence as another objectve by takng nto account the travel cost of the handlng equpment. However, the contaner rehandle and the redundant travel of handlng equpment should be measured by the tme spent for those physcal movements, as ths s far more mportant. Ths study takes nto account the rehandle as an obstructve factor to fast shp handlng wthout quantfyng t, as t s not easy to measure the tme assocated wth those movements, whch s beyond the scope of ths study. Ths paper s organzed as follows. The next secton revews the related lterature. In the thrd secton the proposed algorthm s descrbed. In the subsequent secton, a varety of numercal experments are carred out and presented, and the fnal secton reports the paper s fndngs and conclusons. 2. Lterature revew Stowage plannng s a category of the loadng problem, whch s well recognzed n the lterature and has become wdely used n a varety of transportaton operatons. Most of the work has been done for the Bn Packng Problem. Some of the studes formulate the problem as a 0-1 Mxed Integer Programmng. They nclude the consderaton of multple carton szes, and carton orentatons. Other papers propose computer-based heurstcs. An mportant consderaton for loadng s balance. Martn-Vega (1985) and Amouny et al. (1992) developed a heurstc motvated by the problem of loadng arcrafts or trucks: pack blocks nto a bn so that the center of gravty s as close as possble to a target pont. Mathur (1998) presented an effcent algorthm for a one-dmensonal loadng problem. The goal s to pack homogeneous blocks of gven length and weght n a contaner n such a way that the center of gravty of the packed blocks s as close to a desred pont as possble. The algorthm they proposed s based on the approxmaton of ths problem as a Knapsack Problem, whch s the problem of fttng nto a sack of predefned maxmum weght, tems of dfferent weghts and 4

6 dfferent utltes so as to maxmze the total utlty of the sack. The loadng problem assocated wth arcraft bascally rases no rehandle ssues due to ts storage space characterstcs. The contanershp stowage and load-plannng problem ths paper addresses refers to the arrangement of contaners nsde the shp. Ths s much more dffcult to solve than the arcraft and truck loadng problems due to the fact that the shp s stowage plan has to consder the assgnment of contaners to a three-dmensonal storage space n addton to the restrctons mposed n retrevng contaners from the stacks n the feld. Although ths problem s of hgh mportance to the practtoners, few studes have been conducted on the contaner stowage and load plannng. One of the early works on ths problem was the one conducted by Ima and Mk (1989) who consdered the maxmzaton of GM and the mnmzaton of the loadng-related rehandle when loadng contaners onto one of the shp holds. For smplcty n the soluton process, they formulated the problem as a two-objectve assgnment problem, employng the estmated number of rehandles n the objectve functon nstead of the exact one. The precse number of rehandles s obtaned from the resultng soluton. Ima et al. (2001) followed another approach n order for the exact number of rehandles to be taken nto account n the problem. They formulated the stowage problem as a GM objectve assgnment problem for dentfyng a nonnferor soluton set n terms of the GM and number of rehandles. Multple solutons of the assgnment problem were enumerated, thus computng the exact number of rehandles based on each enumerated soluton. Ths approach, however, generated enormous multple solutons for nearly the same range of a nonnferor soluton set as the method employed n Ima and Mk (1989). Surprsngly, the former took 6000 tmes longer computaton tme than the latter. Ima et al. (2002) modfed the problem only for fndng non-nferor solutons wth acceptable GM. Avrel and Penn (1993) and Avrel et al. (1998) addressed a stowage problem that only mnmzed the unloadng-related rehandles wthout any consderaton for shp s stablty. They formulated the problem as a 0-1 Integer Programmng and appled t for loadng onto a sngle hold lke Ima and Mk (1989). Furthermore, Avrel et al. (2000) developed some characterstcs n the relatonshp between stowage plannng and the colorng of crcle graphs. Dubrovsky et al. (2002) mplemented a GA-based heurstc for the same stowage-plannng 5

7 problem. Todd and Sen (1997) mplemented a GA procedure wth multple crtera such as proxmty n terms of contaner locaton on board and the mnmzaton of unloadng-related rehandle, transverse moment and vertcal moment. Ther study s nterestng because t examned the relatonshp between the rehandle and the shp stablty lke the scope of ths study; however ther stablty s not well defned as used n practce. ote that all the above studes do not assume that each vertcal column n holds contans only contaners of the same destnaton. Wnter (1999) ntroduced the stowage plannng n conjuncton wth load plannng takng nto account the equty of quay crane workload. Ths study also nspred ssues of loadng-related rehandle and shp stablty; however t dd not present any problem formulaton wth these crtera and the relevant soluton method. Martn Jr. et al. (1988) addressed the contaner shp load-plannng problem for the transtaner system. Transtaner operaton s a bottleneck n the loadng process. A heurstc algorthm was developed, based on rules of thumb prevalent n the termnals. The objectves of the heurstc algorthm were the mnmzaton of the transtaner movement tme and the mnmzaton of the number of unloadng-related rehandles. Haghan and Kasar (2001) developed a heurstc algorthm for shp stowage plannng wth the mnmzaton of the contaner handlng cost (actually unloadng-related rehandle), whle keepng the shp s GM acceptable. They developed a heurstc for the problem. Although no other stablty related factors were taken nto account besdes GM n the problem formulaton, those factors such as trm and longtudnal moment were examned n the heurstc. However, these factors were never explctly evaluated n ther solutons. Wlson and Roach (1999, 2000) and Wlson et al. (2001) presented a realstc model, takng nto account all techncal restrctons n order to mplement a commercal usable decson support system. Ther approach was based on decomposng the plannng process nto two phases. In the frst phase, called the strategc process, they created a rough stowage plan, based on groupng the contaners wth the same characterstcs n terms of sze, destnaton, etc., and on assgnng them to blocks of stowage space n the shp. Shp stablty was kept to an acceptable degree by ths assgnment process. These calculatons were performed by a branch and bound procedure. In the second phase, called the tactcal process, ndvdual contaners were assgned to specfc locatons, resultng n a detaled stowage plan. They 6

8 employed a tabu search heurstc for the second phase calculaton. Ther objectves ncluded, among others, the unloadng-related rehandle and shp stablty; however, no detal of the stablty was descrbed n ther study. Due to the complexty of ther soluton methodology, they only show a soluton result for a small sample problem; therefore ts effcency from the practcal vewpont s not shown. More recently, Ambrosno and Scomachen (2004) addressed a stowage-plannng problem wth the objectve of mnmzng the total stowage tme where more practcal constrants are taken nto account such as dfferent types of contaners n length, weght lmt beng accepted for securng shp structure, etc. They assgn some shp holds to contaners wth the same destnaton lke ths study n order to avod unproductve work such as unloadng-related rehandle. However, they do not explctly take nto account loadng-related rehandle. Km et al. (2004) addressed a load-plannng problem wth an objectve of proper arrangement of contaner stacks on board n lght of smooth quay crane operaton and the other of proper contaner retreval sequence from contaner stacks n the yard n lght of smooth transtaner operaton. For ths problem, they developed a beam search algorthm. There have also been some other studes that are related to the contanershp load-plannng problem. As mentoned prevously, delay n contaner handlng at a termnal depends manly on the loadng-related rehandle. Only Ima and Mk (1989) and Ima et al. (2002) address stowage plannng, whle takng nto account the GM, for the reducton of the loadng-related rehandle. Another approach for mnmzaton of the delay s to arrange contaner storage locaton n yard stacks of arrvng export contaners so that unproductve rehandle s mnmzed for a gven contaner shp stowage plan. Km et al. (2000) proposed a dynamc programmng model to determne the storage locaton so that the number of rehandles s mnmzed. The rehandle s also related to the storage space utlzaton on the yard. Taleb-Ibrahm et al. (1993) tackled ths ssue by usng an analytcal model. Km and Km (1994) treated a smlar ssue but wth a Mxed Integer Programmng. All n all, no research work has been conducted on the relatonshp between shp stablty (.e., GM, lst, and trm) and the loadng-related rehandle, whch s the scope of ths study. In ths paper we do not take nto account the unloadng-related rehandle, snce ts 7

9 consderaton makes the problem consderably more dffcult to solve whle ts practcalty s dmnshed. In addton to that, ths study s also motvated by the dffculty n determnng a stowage plan n the context of the appearance of mega contaner shps. As descrbed before, these shps call only at a very few selected hubs, makng unlkely the probablty of experencng unloadng-related rehandle, as the stowage spaces on board the shp are separated and dedcated to each specfc port of call. 3. Problem defnton and soluton method Whlst most major contaner termnals use ether of the two handlng systems: transtaner (Ral-Mounted Gantry Crane or Rubber-Tred Gantry Crane) and straddle carrer, the former has been gettng more popular than the latter especally n major termnals wth heavy traffc handled n a relatvely small area such as those n Japan, because except for the case of the automated transtaners used n Europe, the transtaner can handle contaners stacked hgher n the yard than the straddle carrer system. Therefore, throughout ths secton we consder the transtaner system for the model descrpton, however t s easly adaptable to the straddle carrer system wthout any major change. In the transtaner system, storage space s portoned nto multple blocks, two of whch are shown n Fg Fg. 1 about here The cellular (or LOLO) type contaner shp s consdered n ths study. Fg. 2 shows a typcal cross-sectonal vew of a cellular shp. Each cell n the fgure represents a slot where a contaner can be placed and the number n the cell mples the typcal order of the loadng process. Thus, the order of the loadng process defnes the vertcal and horzontal locatons (and longtudnal locatons as well due to the multple holds onboard) of contaners beng stowed n the shp hold. Ths prncple combnes the loadng-sequence plannng and the stowage plannng. Usually the stowage plannng s separated from the loadng-sequence plannng. However, most Japanese shppng lnes buld a stowage plan n conjuncton wth loadng sequence n order to reduce unproductve rehandle. Those shppng lnes load 8

10 contaners onto a shp n a regular sequence lke Fg. 2 because of possble human errors n loadng tasks resultng from random loadng. From the above dscusson, the stowage plannng combned wth loadng sequence seems a reasonable assumpton, whle n realty the loadng sequence s planned a bt flexbly when a lot of rehandles are expected Fg. 2 about here In ths study, we assume that all contaners are stowed n shp holds, but not on the upper deck. However, the model developed n ths study s adaptable for the case wth contaners stowed both n holds and on the upper deck (or above hatch covers) wthout major modfcatons, f t s assumed that the contaners above the hatch cover of a hold have the same destnaton port as the contaners under the hatch cover. In addton, the model s applcable, wthout modfcatons, to the hatch-cover-less shps wth contaners above the upper deck. ote that t s also assumed that when a hold s not fully loaded, the contaners are stowed wth the top row beng as flat as possble. In the subsequent subsectons, the stablty related evaluaton factors are descrbed. We assume that each contaner has the same center of gravty,.e. the weght s mposed at the center of the contaner along the three axes, even though the center of gravty of a partcular contaner depends on ts overall weght and mass dstrbuton. However, the assumpton made s consdered vald as most contaners are full of small packages contanng general merchandse and have ther overall center of gravty at ther mddle locaton Stablty-related factors Shp stablty s evaluated by three factors: GM, lst and trm (see Derrett (1999) for detals). Stablty ssues rased by lst and trm are tractable by usng the shp s ballast tanks, although they are normally adjusted wthn an acceptable range wthout usng the ballast snce the ballast s reserved for emergent ncdents such as unexpected over-heavy cargoes to be loaded n the subsequent callng ports. Among others the most mportant factor s the GM (more precsely the GM of the shp wth loaded cargoes), whch s the dstance between the center of gravty (G) and the 9

11 metacenter (M) as shown n Fg. 3() and calculated by the followng equaton. GM = G M + w lh 0 (1) T Where lh s the vertcal dstance between the centers of gravty of the shp and contaner, w s the weght of contaner, S s the shp s dsplacement wthout cargo, T s the shp s dsplacement after contaners are loaded( T = S + w )and G 0 M s the dstance between the center of gravty of the shp (G 0 ) and the metacenter (M) Fg. 3 about here The lst, as shown n Fg. 3(), caused by contaners beng loaded onto the shp s measured by tan θ that s calculated by the followng equaton. tan θ = w lw T GM (2) where lw s the horzontal dstance between the vertcal center of the shp and the center of gravty of contaner. As shown n Fg. 3(), the trm (defned as the total of change of drafts forward and aft) s gven by Eq. (3) t= L T w ll GM L (3) Where ll s horzontal length between the center of floataton and the center of gravty of contaner, L s shp s length and and the longtudnal metacenter (M L ). GM L s the dstance between the center of gravty (G) 10

12 Accordng to the Fg. 3(), the metacenter s located very hgh when the trm occurs; therefore, practcally GM L BM L s assumed, where L dstance between the center of buoyancy (B) and M L. BM defned by Eq. (4) s the BM L 3 W L = 12 T (4) where W s the shp s wdth. Assumng the shp s a box, the trm s expressed by Eq. (5), whch s obtaned by nsertng Eq. (4) nto (3). t = 12w 2 W L ll (5) 3.2. Stablty estmaton Based on the above defnton of shp s stablty factors, we formulate the stablty related parameters beng used n objectve functons for the stowage problem. As the GM s gven by Eq. (1), the value defned by Eq. (6) s added to the M G 0 when a contaner at poston of contaner stacks on the yard s loaded (n other words, that contaner s retreved from poston of the stacks) as the j th contaner n the whole loadng sequence, whch s stowed n correspondng poston of j of a shp hold (or a shp bay) as defned n Fg. 2. w lh s j + w (6) As the loadng sequence numbers correspond to postons of contaners onboard the shp, only w lh j depends on the contaner locaton onboard. We defne ths varyng value n the GM by G j = w lh. (7) j 11

13 By usng ths defnton we may formulate the problem only wth the maxmzaton of the GM as follows: [PG] Maxmze = 1 j= 1 G j x j (8) subject to x = 1 (9) j= 1 = 1 j x = 1 j (10) j { 0,1} x =, j (11) j where x j = 1 f a contaner at poston of contaner stacks on the yard s loaded n poston j of shp hold as the j th contaner n the loadng sequence; =0, otherwse and s the number of contaners to be loaded In the formulaton, constrants (9) and (10) ensure that every contaner s loaded wth any order of loadng sequence. As seen n Eq. (2), the lst defnton ncludes the GM. The mnmzaton of the lst requres the maxmzaton of the GM, that s another objectve n the stowage plannng. Ths enables us to only mnmze w lw for the lst objectve. The followng value (hereafter called lst contrbuton) s added to the lst objectve when a contaner at poston s loaded as the j th contaner. H j = w lw (12) j ote that the lst value s ether negatve or postve. The exact lst objectve s the mnmzaton of the absolute value of the lst. Assumng GM>0, we may formulate the problem only wth the mnmzaton of the lst as follows: 12

14 [PH] Mnmze H j x j (13) = 1 j= 1 subject to (9)-(11) In order to make ths formulaton solvable as a mathematcal programmng, we ntroduce Eqs. (15) and (16), resultng n the formulaton [PH ]. [PH ] Mnmze h + + h (14) subject to (9)-(11) = 1 j= 1 H j x j = h + h (15) + h 0, h 0 (16) In accordance wth the trm defnton (5), the value, w ll, s added to the trm j objectve when a contaner at poston s loaded as the j th contaner. Defnng trm contrbuton as T j = w ll (17) j lke the lst, we may, therefore, formulate the problem only wth the mnmzaton of the trm as follows: [PT] Mnmze T (18) j j x j subject to (9)-(11) Lke the lst, the trm formulaton can be rewrtten as follows: [PT ] Mnmze t + + t (19) subject to (9)-(11) = 1 j= 1 T j x j = t + t (20) 13

15 + t 0, t 0 (21) 3.3. Contaner rehandle estmaton Lke Ima and Mk (1989) and Ima et al. (2002), we utlze the estmated number of rehandles n order to take the rehandle objectve nto account n the formulaton. As descrbed n the relevant lterature for contaner rehandle (de Castlho and Daganzo, 1993; Km, 1994; 1997), the dffculty n estmatng the number of contaner rehandles s caused by the random retreve. Ths s typcal for mport contaner dstrbuton; however, t s also the case for export contaner loadng. When loadng contaners, obvously the loadng sequence s predetermned and ths mples that contaner retreve s programmed and not random. Consequently the number of rehandles can be calculated exactly. However, the exact calculaton s based on a predetermned loadng sequence avalable. Because of such a problem nature, t s qute dffcult or even mpossble to formulate the problem as a mathematcal programmng model, wth evaluaton of the exact number of rehandles. We alternatvely ntroduce the dea of probablty, n other words, the estmated number of rehandles to be examned, so that we formulate the problem as a mathematcal programmng model. In Ima and Mk (2002), the rehandle s estmated based on the expected number of rehandles when retrevng each contaner n the block as the frst one to be taken. Wth an assumpton that contaner locatons n a row of the yard are gven the seral number, we let S j be the expected number of rehandles to wthdraw a contaner of locaton as the j th contaner. When wthdrawng a target contaner (black box) n Fg. 4, we obtan the expected number of the hatched contaners to be rehandled. Lettng be the number of contaners n j 1 the row, a set of j 1 contaners s retreved wth the probablty of before another 1 contaner s loaded as the j th one. Thus, the probablty that at least one of the j 1 contaners s not retreved, s j 1 1. (22) 1 14

16 Therefore, the expected number of contaners remanng above the j th retreved contaner (.e., hatched contaner n Fg. 4) before retreval of the j th contaner, could be defned as the multplcaton of the probablty of Eq. (22) and the number of contaners above the j th contaner (correspondng to a contaner of locaton ): S j 1 = 1, (23) 1 j B where B s the number of contaners to be rehandled when a contaner at locaton (black box n Fg. 4) s pcked up as the frst contaner n the loadng sequence. ote that the fgure of B s the one as of the state that the frst retreval takes place for the relevant contaner row. When we envsage contaner retreval from the row, t s ntutonally recognzed that a fewer number of rehandles s assocated wth a specfc contaner retreval f that contaner n the row s retreved late (whch means large value for j ) because of the fact that blockng contaners are more lkely retreved before that contaner. Also, when a contaner s the j th contaner to be retreved, fewer rehandles are expected f fewer contaners are ntally pled as a row. These ntutons justfy the Eq. (23). For nstance, n Fg. 4 the estmated number s 3.86 f the black contaner s retreved as the second (.e., j =2) wth =30, whle t s 3.45 f the black contaner s retreved wth j =5. If the row has =15 contaners wth the same heght as before, the estmated number assocated wth j =2 s Fg. 4 about here S j s a farly good estmaton for the observed number of rehandles wth gven loadng sequences n terms of the total number of rehandles over an entre loadng operaton, accordng to experments n Ima et al. (2002). Whlst n ther study, the number s overestmated by the regresson model they developed, there s a strongly lnear postve assocaton between the estmated and observed numbers; therefore the estmated number s useful n the mnmzaton problem. The optmal soluton to the formulaton wth the 15

17 estmated number of rehandles s not the optmal n terms of the observed (or exact) number of rehandles, however the resultng soluton s consdered a good approxmate one because of the close assocaton between the estmated and exact numbers as mentoned above. Of course, the observed number for the obtaned soluton s computed based on the loadng sequence provded by the soluton. We compute the observed number of rehandles assocated wth a resultng soluton, assumng that rehandled contaners are moved back to the orgnal locatons (actually lower locatons because of target contaners) after pckng up the target contaners. We do not explctly consder the places where the rehandled contaners are temporarly stored whle the target contaners are processed. In the transtaner system they are stored n empty locatons n the same yard bay n practce, whle n the straddle carrer system they are stored n empty locatons n the same sngle row. If no space s avalable for rehandlng (whle t s not lkely because some spare place s reserved n practce for smooth rehandlng), adjacent yard bays are used n the transtaner system and another place s used (for nstance, the next row or reserved space n the yard) n the straddle carrer system. Unlke Ima et al. (2002), ths study assumes multple stack rows on the yard and multple bays onboard beng nvolved n the contaner loadng sequence. However, the entre contaner block splt over several rows on the yard can be arranged as a sngle long row, makng Eq. (23) applcable to ths study, whle needs to be redefned for the arranged sngle row. The rehandle objectve formulaton, then, follows:. [PR] Maxmze = 1 j= 1 S j x j (24) subject to (9)-(11), where the objectve s the mnmzaton of the estmated number of rehandles Formulaton Although the desrable GM s n general one meter, other GM values are used when 16

18 takng nto account other shp condton related factors. Furthermore, loadng planners and shp offcers n charge of cargo handlng may soften the GM and other stablty restrctons n order to reduce the number of requred contaner rehandles that prevent the quck shp turnaround. Such a trade-off analyss requres the set of nonnferor solutons for our problem wth multple objectves. Among a number of technques for generatng a nonnferor soluton set, we employ the weghtng method (Cohon, 1978). In ths method we defne the problem as a mathematcal programmng model wth a sngle objectve that ncorporates multple objectves. Puttng the four objectves nto a sngle objectve wth weghts, we obtan the followng formulaton: + + [PA] Mnmze Z = α G x + β S x + χ( h + h ) + δ( t + t ) j j = 1 j= 1 = 1 j= 1 j j (25) subject to (9)-(11), (15), (16), (20) and (21), where α, β, χ and δ are weghts for the GM, rehandle, lst and trm, respectvely. ote that α s set negatve because of the maxmzaton of the GM Formulaton for unbalanced ntal settng The formulaton of [PA] assumes a contaner-loadng scheme wth the shp beng empty. Ths s the realty n shuttle transportaton between two ports of call. However, n most cases a shp has an tnerary wth callng at more than two ports, where some contaners are already left n shp holds n advance of loadng contaners at a partcular port. Such an ntal stuaton may cause the shp to be nclned. For ths, we extend [PA] to reformulate t as follows: [PA ] Mnmze Z (25) subject to (9)-(11), (16), (21), y jk L k j= 1 jx j y jk H k,k (26) 17

19 = 1 j= 1 = 1 j= 1 H T j j x x j j SH = h h (27) + ST = t t (28) Where SH s the cumulatve value of lst contrbuton of contaners left onboard, ST s the cumulatve value of trm contrbuton of contaners left onboard, y jk = 1 f locaton j of the shp hold belongs to vertcal column k and = 0 otherwse, L k s the lower bound of storage locaton range (or equvalent n loadng sequence number) appled for vertcal column k and H k s the upper bound of storage locaton range (or equvalent n loadng sequence number) appled for vertcal column k. Snce some contaners already exst n each vertcal column, contaners to be loaded at the port of concern are to be stored on the top of them. L k and H k defne the range of locaton for stowage of them n each column. Consequently constrant set (26) assures that contaners are stowed n the range. In constrant sets (27) and (28), SH and ST are the exstng factors n lst and trm before loadng at the port of concern Soluton procedure usng the genetc algorthm As there s no polynomally-bounded tme algorthm beng found for [PA] (and [PA ] as well), we develop a heurstc algorthm by usng the genetc algorthm (GA). ote that whle all the solutons dentfed by the weghtng method are nonnferor, the solutons n ths study are not necessarly nonnferor due to the estmated number of rehandle beng evaluated; therefore the set of nonnferor solutons are reconstructed by computng the observed number of rehandles from the resultng solutons. As GAs are wdely appled for plenty of practcal problems of mathematcal programmng, whch are dffcult to solve n terms of polynomally-bounded computatonal tme, we do not explan the GAs n detal. The stowage-plannng problems are mnmzaton problems; thus, the smaller the objectve functon value s, the hgher the ftness value must be. Havng consdered some alternatve ftness functons, we employed the sgmod functon 18

20 as defned n (29) where y (x) denotes the objectve functon value: f ( x) = 1/(1+ exp( y( x) / 20000)) (29) For dversty n ftness value between ndvduals, the sgmod functon should be appled wth x rangng 2.0 to 2.0. Consequently, takng nto account the objectve functon values n the experments that are descrbed n the next secton, the sgmod functon has been defned as above. The mutaton rate was set to 0.4, based on our prelmnary experments. We apply the tournamentng process, whch Ahuja et al. (2000) proposed for a better soluton. One can apply a GA many tmes startng wth dfferent populatons and choose the best ndvdual obtaned among all the runs. In order to save substantal runnng tme, as an alternatve they take the fnal populaton of two dfferent runs, keep best 50% of the ndvduals n the two runs, and apply the GA agan wth ths mxed populaton as the ntal populaton. 4. umercal experments 4.1. Prelmnary analyss The soluton procedures were coded n C++ language on a Sun SPARC-64G workstaton. Problems used n the experments were generated randomly, but systematcally. We frst tred to compare solutons by GAs wth and wthout tournamentng. The GA wth tournamentng outperformed n soluton qualty as expected, whlst ts computaton tme s 7 tmes longer than the tme wthout tournamentng. We next examned the soluton qualty n detal for fve selected cases of loadng 504 contaners onto a shp wth the capacty of that quantty. Table 1 demonstrates typcal solutons for the fve cases and the total computaton tmes for obtanng a nonnferor soluton set. The stowage plan demonstrates a balanced stowage n terms of weght dstrbuton, as the soluton for case 4 s shown n Fg. 5 where darker boxes represent heaver contaners Table 1 and Fg. 5 about here Prmary analyss 19

21 We set 22 cases wth dfferent shp szes, handlng volumes, contaner stack arrangements, ntal shp condtons, and shp hold arrangements as shown n Table 2. When contaner stacks are grouped by weght (abbrevated by W), they are segregated nto three weght levels. For the shp hold arrangement, arranged by destnaton (D) means that each hold s allocated solely to contaners for a partcular destnaton. In the small shp cases, four dfferent destnatons are assumed, each allocated to two shp holds; whlst n large shp cases three destnatons are consdered, each allocated to one hold, where the total of three holds are taken among others for loadng. For each case, we prepare fve dfferent contaner stack arrangements (abbrevated as SA n subsequent fgures for the results of the experments) wth unform random numbers Table 2 about here The sets of vared weght beng employed for the experments are shown n Table 3. The polcy for settng these weghts follows: weghts for GM (e) and rehandle (f) vary by 20, whle holdng e + f = 100. Gven a set of e and f, weghts for the lst (g) and trm (h) range from 0 to 30 by 15. In accordance wth the prelmnary experments, t s found that these weght sets produced so small amount of dversty for the lst and trm that all four weghts were adjusted as follows for the resultng weghts α, β, χ and δ : α = e 0.001, β = f 0.700, χ = g 0.001, δ = h Table 3 about here Small shp cases We frst examne cases 1 and 2,.e. cases wth random shp hold arrangement. Fg. 6 llustrates the set of nonnferor solutons for case 1. Due to dffculty n representaton of the nonnferor set, four evaluated objectve values assocated wth each soluton are plotted n 20

22 ncreasng order of soluton numbers that are gven wth ncreasng GM value (n meters). The rehandle value here s the observed one beng calculated based on a resultng soluton. The lst (n tanθ ) and trm (n meters) are gven as absolute values n ths and subsequent fgures. The GM s nversely proportonal to the rehandle, whle the lst and trm are confned to null but wth lttle dversty for the trm. Although very small GM value s yelded for solutons 1-9, most solutons seem reasonable, havng GM confned to the range of m. Ths s also the trend for the experment of case Fg. 6 about here For cases 3 and 4 where shp holds are separately used for dedcated destnatons, the overall trend s the same as for cases 1 and 2 but wth many more rehandles. Such an ncrease n rehandle s caused by the shp hold arrangement for these cases. In contrast, contaners can be lkely stowed n any hold for the cases 1 and 2 wth random hold arrangement, resultng n much fewer rehandles. For case 5 where separated contaner stacks on the yard are dedcated for specfc destnatons, fewer rehandles are observed compared to cases 3 and 4, because a group of contaners at a partcular stack are moved as a whole to ts dedcated shp hold. In the cases that some contaners have already been stowed aboard the shp before loadng and the shp leans as a result (cases 6 and 7), the overall trends for soluton are the same as case Large shp cases For cases 8-13 wth the random shp hold arrangement, those results have overall the same trend as case 1. When the shp s ntally nclned (cases 10 and 11), the resultng solutons have consderable dversty n the lst as shown n Fg. 7 for case 10. Solutons when the shp s trm s by the stern, are almost the same as those wth level and nclned condtons Fg. 7 about here ext we examne the cases wth the shp hold beng arranged by destnaton (cases 21

23 14-22). The same trend exsts for every set of cases regardless of the ntal shp condton. Wth any shp condton, a typcal observaton for all the cases, except for the one wth the yard stack arrangement by destnaton, s the enormous number of rehandles, as shown n Fg. 8 for case 14. Ths s caused by a strct restrcton to the order of retrevng contaners n the stacks due to the shp hold arrangement. In contrast, the case wth the yard stack arrangement by destnaton (case 16) yelds hardly any rehandle. In ths case, a set of contaners for a specfc destnaton s ntensvely located n a partcular stack and vrtually retreved by the lump to a partcular shp hold. For all these cases, the lst s nearly null and the trm s not null unlke other cases for large shps. The trend of the lst and trm may be explaned for the same reason Fg. 8 about here Soluton mprovement by reassgnment of the order of the loadng sequence As descrbed prevously, a number of rehandles were observed when shp holds were arranged by destnaton. Ths s because all experments assumed only one quay crane beng employed for the loadng tasks of a specfc shp. ormally bg shps get two or three cranes assgned to them that work smultaneously for speedy loadng/unloadng, resultng n fewer rehandles. For such a mult-crane loadng, the same set of order of the loadng sequence must be gven for each subset of contaners handled by a partcular quay crane; however ths premse does not lead the formulatons [PA] and [PA ]. In order to facltate the formulaton, we assgned the order of the loadng sequence to contaner locatons from the frst bay to the last bay onboard. Whle other strateges n assgnng the order to the locaton onboard can be thought of, they do not affect the resultng solutons n terms of shp stablty,.e., the GM, lst and trm, because they are computed based not on the sequence order but on the locaton onboard. Ths nsght encourages us to apply a dfferent assgnment scheme of the sequence order to the locaton. We assume that multple cranes engage loadng tasks of multple holds. Due to the unque sequence order to a specfc locaton, we also assume that one crane handles ts frst contaner earler than another crane handles ts frst contaner. Based on ths 22

24 premse, we may arrange the sequence order such that the frst contaner (or the locaton onboard beng treated frst) handled by crane 1 corresponds to the frst one, the frst by crane 2 s the second one, the second by crane 1 s the thrd, the second by crane 2 s the fourth, etc. By these recalculatons, hardly any rehandle s observed even for those cases that yelded a lot of rehandles n the prevous experments. One such example s shown n Fg. 9 for case Fg. 9 about here Concludng remarks Ths paper addressed the problem of obtanng a nonnferor soluton set for the contaner shp stowage plannng. For the shp loadng tasks, a major concern s shp stablty, typcally the GM, lst and trm. Another concern s contaner rehandlng whch occurs when specfc contaners are pcked up from the contaner stacks on the yard. The problem was defned as a mult-objectve nteger programmng, for whch we obtaned a set of nonnferor solutons by usng the weghtng method. A wde varety of experments demonstrated that the solutons by ths formulaton were acceptable for practcal use when no rehandle takes place n unloadng process. Whle we appled the GA wth the tournament for better soluton, ts computaton tme s larger than one wthout the tournament. If termnal operators requre faster plannng when our approach s mplemented, they may use the algorthm wthout the tournament. References Ahuja, R.K., Orln, J.B., Twar., A., A greedy genetc algorthm for the quadratc assgnment problem. Computers & Operatons Research 27, Ambrosno, D., Scomachen, A., Tanfan., E., Stowng a contanershp: the master bay plan problem. Transportaton Research A 38, Amouny, S. V., Barthold, III, J.J., Vande Vate, J.H., Zhang, J., Balanced Loadng. Operatons Research 40, Avrel, M, Penn, M., Exact and approxmate solutons of the contaner shp stowage 23

25 problem. Computers & Industral Engneerng 25, Avrel, M, Penn, M., Shprer,., Contaner shp stowage problem: complexty and connecton to the colorng of crcle graphs. Dscrete Appled Mathematcs 103, Avrel, M, Penn, M., Shprer,., Wtteboon, S., Stowage plannng for contaner shps to reduce the number of shfts. Annals of Operatons Research 76, de Castlho, B., Daganzo, C.F., Handlng strateges for mport contaners at marne termnals. Transportaton Research B 27, Cohon, J.L., Multobjectve Programmng and Plannng, Academc Press, ew York. Derrett, D.R., Shp Stablty for Masters and Mates, Butterworth-Henemann, Oxford. Dubrovsky, O., Levtn, G, Penn M., A genetc algorthm wth a compact soluton encodng for the contaner shp stowage problem. Journal of Heurstcs 8, Haghan, A., Kasar, E.I., A model for desgnng contaner loadng plans for contanershps. Presented n Annual Conference for Transportaton Research Board. Ima, A., Mk, T., A heurstc algorthm wth expected utlty for an optmal sequence of loadng contaners nto a contanerzed shp. Journal of Japan Insttute of avgaton 80, (n Japanese). Ima, A., shmura, E., Sasak, K., Papadmtrou, S., Soluton comparsons of algorthms for the contanershp loadng problem. Proceedngs of the Internatonal Conference on Shppng: Technology and Envronment, avalable n CD-ROM. Ima, A., shmura, E., Papadmtrou, S., Sasak, K., The contanershp loadng problem. Internatonal Journal of Martme Economcs 4, Km, K.H., Analyss of rehandles of transfer crane n a contaner yard. APORS-Conference 3, Km, K.H., Evaluaton of the number of rehandles n contaner yards. Computers & Industral Engneerng 32, Km, K.H., Km, D.Y., Group storage methods at contaner port termnals. The materals Handlng Engneerng Dvson of 75th Annversary Commemoratve Volume ASME, Km, K.H., Park, Y.M., Ryu, K.-R., Dervng decson rules to locate export contaners n contaner yards. European Journal of Operatonal Research 124, Km, K.H., Kang, J.S., Ryu, K.R., A beam search algorthm for the load sequencng of outbound contaners n port contaner termnals. OR Spectrum 26, Martn Jr., G.L., Randhawa, S.U., McDowell, E.D., Computerzed contaner-shp load plannng: a methodology and evaluaton. Computers & Industral Engneerng 14, Martn-Vega, L.A Arcraft load plannng and the computer: descrpton and revew. Computers & Industral Engneerng 9, Mathur, K., An nteger-programmng-based heurstc for the balanced loadng problem. 24

26 Operatons Research Letters 22, Taleb-Ibrahm, M., de Castlho, B., Daganzo, C.F., Storage space vs handlng work n contaner termnals. Transportaton Research B 27, Todd, D.S., Sen, P., A multple crtera genetc algorthm for contanershp loadng. Proceedngs of the Seventh Internatonal Conference on Genetc Algorthms, Wlson, I.D., Roach, P.A., Prncples of combnatoral optmzaton appled to contaner-shp stowage plannng. Journal of Heurstcs 5, Wlson, I.D., Roach, P.A., Contaner stowage plannng: a methodology for generatng computersed solutons. Journal of Operatonal Research Socety 51, Wlson, I.D., Roach, P.A., Ware, J.A., Contaner stowage pre-plannng: usng search to generate solutons, a case study. Knowledge-Based Systems 14, Wnter, T., Onlne and Real-Tme Dspatchng Problems, PhD thess, Techncal Unversty of Braunschweg, Germany. 25

27 Table 1. Soluton profle for cases of loadng 504 contaners Case Soluton CPU tme GM(m) Rehandle Heel( tan θ ) Trm(m) (s)

28 Table 2. Computatonal cases Case # Shp sze Contaner volume Stack arrang. Intal shp cond. Shp hold arrang. 1 S M R L R 2 S M W L R 3 S M R L D 4 S M W L D 5 S M D L D 6 S F R H R 7 S F R T R 8 L M R L R 9 L M W L R 10 L M R H R 11 L M W H R 12 L M R T R 13 L M W T R 14 L M R L D 15 L M W L D 16 L M D L D 17 L M R H D 18 L M W H D 19 L M D H D 20 L M R T D 21 L M W T D 22 L M D T D Keys Shp sze: S - capacty of 504 TEUs, L - capacty of 2016 TEUs Contaner volume: F TEUs, M 504 TEUs Stack arrangement: R random, W grouped by weght, D grouped by destnaton Intal shp condton: L level, o H wth lst ( θ = 10 and respectvely), o 15 starboard sde for shp sze=s and L, T wth trm (0.5m by the stern for both shp szes) Shp hold arrangement: R random, D arranged by destnaton 27

29 Table 3. Weght sets # e f g h same as sets 2 to 17 but wth g= same as sets 8 to 17 but wth g=

30 BAY Fg. 1. Yard layout 29

31 Fg. 2. Cross secton of cellular contaner shp 30

32 M L water lne M G θ water lne t B G () GM and lst () Trm Fg. 3. Stablty factors 31

33 Fg. 4. Contaner stack n a yard 32

34 BAY 1 BAY 2 BAY 3 BAY 4 BAY 5 BAY 6 BAY 7 BAY 8 Fg. 5. Cross sectonal vew of contaners on board 33

35 GM Soluton # SA=1 SA=2 SA=3 SA=4 SA=5 Trm Soluton # SA=1 SA=2 SA=3 SA=4 SA= Lst Soluton # SA=1 SA=2 SA=3 SA=4 SA=5 Rehandle Soluton # SA=1 SA=2 SA=3 SA=4 SA=5 Fg. 6. onnferor soluton sets for case 1 34

36 GM SA=1 SA=2 SA=3 SA=4 SA=5 Trm SA=1 SA=2 SA=3 SA=4 SA= Soluton # Soluton # Lst Soluton # SA=1 SA=2 SA=3 SA=4 SA=5 Rehandle Soluton # SA=1 SA=2 SA=3 SA=4 SA=5 Fg. 7. onnferor soluton sets for case 10 35

37 GM SA=1 SA=2 SA=3 SA=4 SA=5 Trm SA=1 SA=2 SA=3 SA=4 SA= Soluton # Soluton # L s t SA=1 SA=2 SA=3 SA=4 SA=5 Rehandle SA=1 SA=2 SA=3 SA=4 SA= Soluton # Soluton # Fg. 8. onnferor soluton sets for case 14 36

38 GM SA=1 SA=2 SA=3 SA=4 SA=5 Trm SA=1 SA=2 SA=3 SA=4 SA= Soluton # Soluton # Lst SA=1 SA=2 SA=3 SA=4 SA=5 Rehandle SA=1 SA=2 SA=3 SA=4 SA= Soluton # Soluton # Fg. 9. Improved nonnferor soluton sets for case 14 37