Optimization on Retrieving Containers Based on Multi-phase Hybrid Dynamic Programming

Transcription

1 Available olie at ScieceDirect Procedia - Social ad Behavioral Scie ce s 96 ( 2013 ) Abstract 13th COTA Iteratioal Coferece of Trasportatio Professioals (CICTP 2013) Optimizatio o Retrievig Cotaiers Based o Multi-phase Hybrid Dyamic Programmig Zha BIAN*, Zhi-hog JIN Trasport Maagemet College, Dalia Maritime Uiversity, Dalia, , P R Chia Retrievig export cotaiers from a cotaier yard is a importat part of the ship-loadig process. This paper presets a three-phase hybrid algorithm to solve for a optimized worig pla for a gatry crae to retrieve all the cotaiers from a give yard accordig to a give order. The optimizatio goal is to miimize the umber of cotaier movemets, as well as obtais several alterative retrievig sequeces through various methods. With a etwor, phase three costructs a shortest path problem ad derives the optimal sequece by dyamic programmig. Numerical testig results show that the algorithm is able to solve istaces with more tha 2000 cotaiers, which is withi the rage of real-world applicatios. Moreover, the umber of movemets approaches the lower boud i most cases, ad the resultig retrievig sequece is efficiet The Authors. Published by Elsevier by Elsevier Ltd. Ope B.V. access uder CC BY-NC-ND licese. Selectio ad/or peer-review peer-review uder resposibility uder resposibility of Chiese of Overseas Chiese Trasportatio Overseas Trasportatio Associatio (COTA). Associatio (COTA). Keywords: cotaiers; retrievig sequece; heuristic rules; dyamic programmig; three-phase hybrid algorithm 1. Itroductio Over the past 20 years, cotaier termials have witessed the icreased world-wide flow of cotaiers ad more popular larger-sized cotaier vessels. The competitio amog termials has become promiet which maes the efficiecy of port operatio a importat factor i succeedig i the fierce competitio. Of all of the popular service performace measures, vessel turaroud time, which is the average time that a vessel stays i a termial, is the most importat. Port operatios ca be geerally divided ito two parts: the dischargig operatio durig which cotaiers are uloaded from cotaierships, ad the loadig operatio durig which cotaiers are loaded oto ships. I most cotaier termials, a large portio of the turaroud time of a vessel is cosumed by the two processes. I this paper, we study the problem of retrievig cotaiers from a yard i a give sequece, * Correspodig author. Tel.: address: biazha1990@163.com The Authors. Published by Elsevier Ltd. Ope access uder CC BY-NC-ND licese. Selectio ad peer-review uder resposibility of Chiese Overseas Trasportatio Associatio (COTA). doi: /j.sbspro

2 Zha Bia ad Zhi-hog Ji / Procedia - Social ad Behavioral Scieces 96 ( 2013 ) which is a importat part of the ship-loadig process. Most yards stac up cotaiers to utilize more ad more precious space. Ulie may usual storage systems that are capable of providig radom access to all stored items, oly those located at the top are directly accessible to the yard craes. I additio, cotaiers have to be loaded oto ships accordig to the stowage pla, which specifies the locatio of each cotaier o the ship, ad thus largely determies the order the cotaiers are to be loaded oto the vessel. Extra movemets, that waste time ad moey, occur whe a cotaier is due to be retrieved from the yard but is buried beeath other oes. Oe potetial way to reduce relocatios ad loadig time is to pre-marshal the export cotaiers before loadig starts. Lee ad Hsu (2007) focused o the cotaier pre-marshalig problem for a sigle bay, ad developed a iteger programmig model that yields a step-by-step worig pla for the crae. Worig i a similar directio, Lee ad Chao (2009) solved much larger istaces with a eighborhood search approach. However, all these models focus o yard pre-marshalig, ad are differet from the problem addressed i the curret wor. The literature related to optimizig the retrieval process i a yard is ot extesive. It was first proposed by Chug et al. (1988), the Kim ad Kim (1997, 1999) formulated a mixed iteger programmig model for the routig problem of a crae loadig export cotaiers out of the stac area oto waitig yard trucs, but the scope of this model is limited to the routig of cotaier carriers. Kim ad Hog (2006) used a brach-ad-boud approach, as well as a simple ad effective heuristic, to determie how cotaiers i a sigle bay ca be retrieved with the least umber of movemets. The largest example preseted i their paper has 30 cotaiers, which is sigificatly smaller tha the typical umber of cotaiers i practice. Also attemptig to miimize re-hadles, Lee ad Lee (2010) proposed a three-phase heuristic ad iteger programmig to solve for a optimized worig pla for a crae. I this paper we develop a hybrid algorithm for the cotaier retrieval problem. Give the iitial layout of a yard with multiple bays, the algorithm obtais a movemet sequece for the crae to retrieve all the cotaiers i a specified order. The optimizatio goal is to miimize the total umber of cotaier movemets, as well as the overall worig time. The algorithm is able to solve istaces with more tha 2000 cotaiers i a few hours, which fits well ito the time widow betwee the arrival of a ship ad the startig of the loadig process. This performace brigs the algorithm withi the rage of practical applicatios. crae trolley spreader Z (tier) a bay X (bay) Y(stac) rail a stac Fig. 1. Illusio of a rail mouted gatry crae retrievig cotaiers amog blocs. The operatig strategy of ay cotaier yard has to be developed accordig to the equipmet used i that particular locatio. I this research, we focus o yards that use rail mouted gatry craes (RMGC) as their major cotaier hadlig equipmet. As illustrated i Fig. 1., the crae is composed of a gatry that moves i the bay dimesio, a trolley that moves i the stac dimesio o top of the gatry, ad a spreader that moves i the tier

3 846 Zha Bia ad Zhi-hog Ji / Procedia - Social ad Behavioral Scieces 96 ( 2013 ) dimesio. Whe a RMGC lifts a cotaier from the yard, the crae first positios itself at the bay where the target cotaier resides, positios its trolley over the target stac, lowers its spreader to hold oto the cotaier, ad the lifts it up. After liftig the cotaier, the RMGC ca lower the cotaier oto a truc waitig at oe ed of the crae, or place the cotaier o the top of aother stac. We assume the RMGC is capable of movig cotaiers betwee differet bays, as ewer models commoly do. We also assume that there is oly oe RMGC i the worig area, ad thus collisio betwee craes is ot a problem. This paper is divided ito five sectios. Followig this itroductio, the secod sectio will defie the cotaier retrieval problem. The mathematical methods will be developed i detail i the third sectio, followed by computatioal examples ad aalysis i sectio four. Fially, coclusios ad future research will be discussed i the fifth sectio. 2. The cotaier retrieval problem Give a iitial layout of a yard, the cotaier retrieval problem yields a movemet sequece that retrieves all the cotaiers from the yard, oe at a time i a specified order, such that the umber of cotaier movemets as well as worig time is miimized. The basic assumptios used i this research are listed below: 1. Cotaiers of differet dimesios are stored i differet stacs. The vast majority of cotaiers are either 20 or 40-feet log. I priciple, cotaiers of differet legths ca be mixed together i oe sigle stac. For example, oe 40-feet cotaier ca rest o two 20-feet cotaiers (but ot the other way roud). However, doig this complicates yard operatios ad is avoided wheever possible i practice. To simplify otatios ad explaatios, we will assume that all the cotaiers are of the same legth. 2. Oly oe ship is presumed to be preset. Although it is possible for large ports to have two or more ships beig loaded at the same time with cotaiers tae from the same yard, served by the same RMGC, we do ot cosider this sceario. 3. The loadig order of the cotaiers is ow. I practice, the loadig sequece of a ship is determied well before loadig starts (more tha 6 h i most cases).throughout the paper, the cotaiers are umbered with cosecutive itegers startig from 1, ad those with smaller umbers have to be retrieved earlier, show as Fig. 1.. Fig. 2(a) shows a bay with three stacs, where all six cotaiers ca be retrieved without additioal re-hadles. I the bay show i Fig. 2(b), cotaiers 1 ca be retrieved directly, but cotaiers 4 ad 6 have to be relocated to other stacs before cotaiers 2 ad 3 ca be retrieved. Therefore, a lower boud for the umber of relocatio movemets eeded to retrieve all six cotaiers i this stac is two, ad the umber of total movemets is at least eight (the total umber of cotaiers plus the miimum umber of relocatios). A lower boud of the total umber of movemets for the etire yard ca be easily estimated by summig up the miimum umber of movemets for all the stacs i the yard. a 1 b Fig. 2. Bays that are retrieved with differet umbers of movemets. Similarly, we umber the stacs with cosecutive itegers startig from 1. For example, if there are 16 stacs i each bay, the the stacs i the first bay are umbered from 1 to 16, the stacs i the ext bay are umbered from 17 to 32, ad so o. Stac 0 represets the truc which pars at oe ed of the crae to carry the retrieved cotaier away. I this research we use a triplet cosisted of the cotaier idetificatio umber, the origiatig

4 Zha Bia ad Zhi-hog Ji / Procedia - Social ad Behavioral Scieces 96 ( 2013 ) stac, ad the operatio type to represet a cotaier movemet. For example, the movemet (x, m, U) represets a liftig operatio of cotaier x from stac m; the movemet (x,, D) represets loadig cotaier x to stac. A movemet sequece is a ordered set of cotaier movemets ad will be cosidered ifeasible whe oe or more followig coflicts occur: 1. The cotaier is ot i the stac while the crae executes a liftig operatio. 2. There are misoverlays whe the crae lifts the target cotaier. 3. The spreader is already fully loaded whe liftig operatio begis. 4. The stac has reached its highest height whe a cotaier is to be loaded ito. 5. The spreader does ot hold the cotaier or the spreader is empty whe executig a loadig operatio. 6. Cotaiers with bigger umbers leave the yard earlier tha those with smaller umbers. A feasible movemet sequece cotais a umber of movemets arraged i a certai order that, whe executed, retrieves all the cotaiers i the preferred order without ecouterig coflicts. (x, m, U) alog with (x,, D) form a complete movemet pair. Thus we defie the umber of pairs as the sequece legth. The quality of a feasible movemet sequece ca be determied by two aspects. Oe is the sequece legth, ad the other is the total time the crae eeds to execute it. The former ca be determied by simple coutig, ad the latter ca be accurately estimated from the performace data of the crae, which specifies the time eeded for the crae to repositio itself betwee movemets, ad the time for the crae to perform the movemets themselves. The optimizatio goal of the algorithm proposed i this research is to derive a optimal movemet sequece with the miimum worig time. 3. The three-phase hybrid algorithm The hybrid algorithm cosists of three phases executed oe after the other, amely the iitial phase that geerates a feasible retrievig sequece with heuristic rules, the secod phase that obtais several alterative retrievig sequeces through various ways ad the third phase that derives a optimal sequece by dyamic programmig. We ext itroduce the three phases i more detail The iitial phase The tas of the iitial phase of the heuristic is to develop a feasible retrievig sequece that eables the crae to retrieve all the cotaiers i the yard without ecouterig coflicts. The followig otatios will be used to describe the heuristic: Notatios N The umber of cotaiers i the iitial layout. S H The umber of stacs i the layout. The maximum height of stacs. a The cotaier with the serial umber, 1,2,..., N. s a The serial umber of the stac where a stays, s a 1,2,..., S. top s The serial umber of the cotaier o the top of stac s, if there is o cotaier i stac s, the top s N 1.

5 848 Zha Bia ad Zhi-hog Ji / Procedia - Social ad Behavioral Scieces 96 ( 2013 ) b top s a The cotaier o the top of stac s a. mi s The miimum serial umber of cotaiers i stac s. U s s S,mi s > b The set of stacs satisfyig the costrait mi s > b. arg max mi i S\ s a i Defie the stac (ay stac except for stac s a ) with maximum mi i as stac. arg mi mi i U i Defie the stac (of the set U ) with the miimum mi i as stac. E s The umber of empty slots i stac, E s 1,2,...,H. OBT a The set of bloced cotaiers to cotaier a. ST The set of bloced cotaiers satisfyig the followig costraits-its serial umber is larger tha b, o the top of stacs (except for stac s ), mi s 6< < mi s ( represets the serial umber of a bloced cotaier). st The cotaier of the set ST with maximum serial umber. I this phase, the heuristic attempts to retrieve all the cotaiers i the required order. If the target cotaier is readily available, it is retrieved ad loaded oto the truc immediately. Otherwise, the cotaiers that bloc the target cotaier are moved to certai stacs by heuristic rules. Fig. 3. illustrates the decisio tree of retrievig sequece. a o top of the stac ot o top of the stac load a directly oto the truc load to aother stac b o mis-overlays occur choose stac Es ( ) 1 ST= Es () 1 ST load b to stac firstly load stto stac, the load b to stac mis-overlays occur load b to stac Fig. 3. The decisio tree of retrievig sequece.

6 Zha Bia ad Zhi-hog Ji / Procedia - Social ad Behavioral Scieces 96 ( 2013 ) As is show i Fig. 3., two differet circumstaces occur whe choosig a temporary stac s for cotaier b : 1. mi s > b, i.e., there is o more misoverlays after cotaier b loaded to stac s. If several stacs satisfy the coditio, choose arg mi mi i. Thus, the serial umber of b will be close to the miimum serial umber of cotaiers i i Ustac, so as to reduce the probability for occupyig the slot for the cotaier of which serial umber is bigger tha that of b. If stac with several empty slots exists, firstly chec whether there are ay ST to utilize the slots, the load ST to stac accordig to descedig order till b is i stac or stac reaches its maximum height. 2. mi( s) b, i.e., more misoverlays will occur after cotaier b loaded to ay stac. Therefore, the stac which has the miimal impact o subsequet operatio after b loaded will be chose. It is obvious that the bigger for the serial umber of overlay cotaier by cotaier b, the later the ext movemet of cotaier b happes. So arg max mi i is chose as the temporary stac for cotaier b. The heuristic rules i S\ sca be illustrated with the bay show i Fig. 4.. The iitial retrievig sequece obtaied is (1,1,U), (1,0,D), (7,3,U), (7,6,D), (6,2,U), (6,6,D), (2,2,U), (2,0,D), (11,5,U), (11,4,D), (3,5,U), (3,0,D), (4,3,U), (4,0,D), (5,2,U), (5,0,D), (6,6,U), (6,0,D), (7,6,U), (7,0,D), (8,6,U), (8,0,D), (12,1,U), (12,2,D), (9,1,U), (9,0,D), (11,4,U), (11,2,D), (10,4,U), (10,0,D), (11,2,U), (11,0,D), (12,2,U), (12,0,D). It is easy to see that i the iitial retrievig sequece, every cotaier with misoverlays (e.g. cotaier 2) is moved at least twice (oce for movig out of the way of the bloced cotaier, ad agai for retrieval), ad every o-bloced cotaier is moved oly oce (e.g. cotaier 1) The secod phase Fig. 4. Illustratio of the iitial yard pla. The secod phase taes the iitial feasible sequece as the iput, ad attempts to fid more alterative movemets while maitaiig feasibility. Basic ideas to geerate alterative sequeces are as follows: 1. Reverse two pairs of cotaier movemets. If there are four differet stacs ivolved i two pairs of cotaier retrievig movemets, we ca chage the order of the two pairs. 2. The iitial stac is replaced by the alterative stac. For istace, there is a sequece composed of (x, m, U), (x,, D),, (x,, U), (x, s, D), which does ot cotai other cotaiers besides cotaier x betwee every two movemets. If stac t is completely ot used i the sequece, ad o more misoverlays occur after loadig cotaier x ito stac t, the sequece ca be replaced by (x, m, U), (x, t, D),, (x, t, U), (x, s, D). 3. Brig movemets forward. This method goes for a cotaier with more tha two pairs of movemets. Assumig that cotaier x ows two pairs of movemets (x, m, U), (x,, D),, (x,, U), (x, s, D), but the whole sequece is (x, m, U), (x,, D),, (y, p, U), (y, q, D), (x,, U), (x, s, D). As stacs use i the pair (x,, U), (x, s, D) are differet from that of (y, p, U), (y, q, D), (x,, U), (x, s, D) ca be moved to the frot of (y, p, U), (y, q, D), ad brought forward till meet a pair with the same stacs or a previous movemet pair for cotaier x (i.e. (x, m, U), (x,, D)). 4. Brig movemets bacward. The method is almost the same as the idea 3, but of opposite searchig directio. 5. Brig a pair forward. This method goes for a cotaier with oly oe pair of movemets which is aimed at loadig the cotaier oto a truc. Assumig that cotaier x oly ows oe pair of movemets (x,, U), (x, 0,

7 850 Zha Bia ad Zhi-hog Ji / Procedia - Social ad Behavioral Scieces 96 ( 2013 ) D), but movemets i frot of these do ot use the same stac. Besides, cotaiers with smaller serial umber have bee retrieved out of the yard. Based o the above coditios, (x,, U), (x, 0, D) ca be moved forward. 6. Brig a pair bacward. The method is almost the same as the idea 5, but of opposite searchig directio. The ideas ca be applied to geerate alterative sequeces for the iitial retrievig sequece metioed i 3.1. Accordig to the idea 2, cotaier 6 ca be retrieved from stac 2 ad loaded to stac 1. Thus, a alterative sequece ca be idicated as(6,2,u), (6,1,D), (2,2,U), (2,0,D), (11,5,U), (11,4,D), (3,5,U), (3,0,D), (4,3,U), (4,0,D), (5,2,U), (5,0,D), (6,1,U), (6,0,D). Accordig to the idea 1, we ca reverse the order of the pairs (8,6,U), (8,0,D) ad (12,1,U), (12,2,D). Therefore, aother alterative sequece ca be represeted as (12,1,U), (12,2,D), (8,6,U), (8,0,D) The third phase Phase three aims to obtai the optimal sequece by the followig steps: firstly, build a etwor without loops for all retrievig movemets to formulate a shortest path problem, ad the use dyamic programmig to solve the problem. Numerous alterative sequeces ca be derived though the six ideas above. We build a shortest path problem based o the etwor of which vertexes ad edges represet the storage coditio ad time-cosumptio of movig ad retrievig operatio respectively. Fig. 5. ad Fig. 6. are etwors of the iitial retrievig path ad the alterative path with two alterative retrievig sequeces (metioed i 3.2) respectively. Vertex illustrates the iitial storage coditio (show as Fig. 4.) ad vertex illustrates the fial yard layout after all retrievig operatios. Parameters used i dyamic programmig which is proposed to solve the shortest path problem are as follows: Parameters N The umber of cotaiers i the iitial layout. m H The umber of bays i the iitial layout. The umber of stacs i a bay. The maximum height of stacs. The total umber of stages, 1,2,...,N. d i The state of stac i after cotaier is retrieved, 1,2,...,N, i 1,2,..., m. s j The state of bay j after cotaier is retrieved, j=,,..., j-1 +1 j-1 +2 j j 1,2,..., m. S The state of the yard after cotaier is retrieved, = 1, 2,..., m, s d d d, 1,2,...,N. S s s s, 1,2,...,N a All the movemets taig place while retrievig cotaier, 1,2,...,N. -1 time a S The time-cosumptio of movig ad retrievig operatios while coductig a uder the state of. 1 S, 1,2,...,N

8 Zha Bia ad Zhi-hog Ji / Procedia - Social ad Behavioral Scieces 96 ( 2013 ) T S The miimal time-cosumptio of movig ad retrievig operatios to retrieve the rest of N cotaiers uder the state of S, 1,2,...,N The problem ca be described as: T S mi time a S T S, where 0 1 S turs ito S via the a 0 a 1 retrieval decisio u S ad the movemet a, i.e., S S. By parity of reasoig, the miimal time 1 0 p p-1 cosumed by the whole process ca be represeted as: T S mi time a S + T S, where p 1 2 p-1 a p a, a, a p =1 S S, p 1,2,...,. Ad the optimal retrieval decisios are u S, u S,..., u S. The secod 1 2 ad the third phase are iterative, ad will ot termiate util cosecutive iteratios caot miimize the RMGC worig time. 1,1,U 1,0,D 7,3,U 7,6,D 6,2,U 6,6,D 2,2,U 2,0,D 11,5,U 11,4,D 3,5,U 3,0,D 4,3,U 4,0,D 5,2,U 5,0,D 6,6,U 6,0,D 7,6,U 7,0,D 8,6,U 8,0,D 12,1,U 12,2,D 9,1,U 9,0,D 11,4,U 11,2, D 10,4,U 10,0, D 11,2,U 11,0,D 12,2,U 12,0,D Fig. 5. Illustratio of the iitial retrievig path. 6,2,U 6,1,D 2,2,U 2,0,D 11,5,U 11,4,D 1,1,U 1,0,D 7,3,U 7,6,D 6,2,U 6,6,D 2,2,U 2,0,D 11,5,U 11,4,D 3,5,U 3,0,D 4,3,U 4,0,D 5,2,U 5,0,D 6,1,U 6,0,D 3,5,U 3,0,D 4,3,U 4,0,D 5,2,U 5,0,D 6,6,U 6,0,D 7,6,U 7,0,D 12,1,U 12,2,D 8,6,U 8,0,D 8,6,U 8,0,D 12,1,U 12,2,D 9,1,U 9,0,D 11,4,U 11,2,D 10,4,U 10,0, D 11,2,U 11,0,D 12,2,U 12,0,D Fig. 6. Illustratio of the alterative retrievig path.

9 852 Zha Bia ad Zhi-hog Ji / Procedia - Social ad Behavioral Scieces 96 ( 2013 ) Computatioal results I this sectio, we provide computatioal examples to demostrate the performace of the algorithm developed i this paper. The proposed algorithm has bee implemeted i Microsoft Visual C++ ad ru o a persoal computer which has a Core I5 CPU ruig at 2.50GHz ad with 4.0GB memory. I all the cases, the cotaiers are geerated, ad radomly placed i the yard, subject to pre- determied umber of bays, stacs per bay, ad maximum stac height. Some preferece settigs are show as follows (the same as Lee ad Lee (2010)): the speed of the gatry is 3.5 s per bay, the speed of the trolley is 1.2 s per cotaier width, the acceleratio ad deceleratio time loss combied for the gatry is 40 s, while the pic-up ad place-dow time combied for the spreader is 30 s. Example 1 presets the case show as Fig. 4. i sectio 3.1. It is a small istace with oly 1 bay cotaiig 6 stacs ad a maximum height of 6. The estimated lower boud for the umber of movemets to retrieve all 12 cotaiers is 16. However, the umber of movemets i sequece is 17 due to oe relocatio operatio of cotaier 11. The istace was solved i s by the proposed algorithm with 502 iteratios while i less tha 0.1s at the iitial phase with oly 1 iteratio. The retrieval wor i this case ca be completed i s by a RMGC via the proposed algorithm while i s at the iitial phase. Next we compare the curret algorithm with the oe proposed by Lee ad Lee (2010). Their heuristic has already preseted 50 istaces (show as Table 1 of Lee ad Lee (2010)) which are also used as experimets here. The result of the 50 istaces is show i Table 1. The curret algorithm resulted i fewer movemets i all 50 istaces, ad the CPU time is sigificatly lower tha Lee & Lee (2010), which maes the curret algorithm of much more practical sigificace. Moreover, the RMGC worig time is also optimized by the curret algorithm. Table 1. Compariso of the result with the heuristic by Lee ad Lee (2010). ID L M CPU T T/M the curret algorithm Lee & Lee (2010) the curret algorithm Lee & Lee (2010) the curret algorithm Lee & Lee (2010) the curret algorithm R011606_0070_ R011606_0070_ R011606_0070_ R011606_0070_ R011606_0070_ R021606_0140_ R021606_0140_ R021606_0140_ R021606_0140_ R021606_0140_ R041606_0280_ R041606_0280_ R041606_0280_ R041606_0280_ Lee & Lee (2010) R041606_0280_

10 Zha Bia ad Zhi-hog Ji / Procedia - Social ad Behavioral Scieces 96 ( 2013 ) R061606_0430_ R061606_0430_ R061606_0430_ R061606_0430_ R061606_0430_ R081606_0570_ R081606_0570_ R081606_0570_ R081606_0570_ R081606_0570_ R101606_0720_ R101606_0720_ R101606_0720_ R101606_0720_ R101606_0720_ R011608_0090_ R011608_0090_ R011608_0090_ R011608_0090_ R011608_0090_ R021608_0190_ R021608_0190_ R021608_0190_ R021608_0190_ R021608_0190_ R041608_0380_ R041608_0380_ R041608_0380_ R041608_0380_ R041608_0380_ R061608_0570_ R061608_0570_ R061608_0570_ R061608_0570_ R061608_0570_ ID: Istace tae from Table 1 of Lee ad Lee (2010). L: Lower boud o the umber of movemets. M: Number of movemets i sequece. CPU: CPU time, i secods. T: Crae worig time.

11 854 Zha Bia ad Zhi-hog Ji / Procedia - Social ad Behavioral Scieces 96 ( 2013 ) T/M: Average time of each movemet. As the heuristic by Lee & Lee (2010) deals with o more tha 10 bays, total 720 cotaiers, i the last part of this sessio, we test the performace of the curret algorithm with larger-sized cases, compared with the iitial phase of the algorithm. Table 2 shows four more examples proposed i this part. For example, R101606_0816_001 presets a large istace with 10 bays, each cotaiig 16 stacs. The maximum height of the stacs is 6, ad there are 816 cotaiers i the yard. Therefore, the space utilizatio rate is 85%. Table 3 shows the result of the curret algorithm ad the iitial phase of it respectively. The iitial phase too oly oe iteratio to get the feasible retrievig sequece while the whole algorithm too much more iteratios, as a result of the secod phase ad the third phase, to get the optimal sequece. Although the umbers of movemets i two solutios are the same, the RMGC worig time of the curret algorithm is reduced dramatically compared with that of the iitial phase. Ad eve though CPU time of the curret algorithm is about 30 times more tha that of the iitial phase, it is still withi acceptable limits for radomly staced yards. Table 2. Additioal umerical examples. ID Bay Stac Max Height NO. of cotaiers Utilizatio R101606_0816_ % R101606_0864_ % R200806_0720_ % R301606_2160_ % Table 3. Compariso of the result with the iitial phase. ID L I M CPU T T/M iitial curret iitial curret iitial curret iitial curret iitial curret R101606_0816_ R101606_0864_ R200806_0720_ R301606_2160_ L, M, CPU, T, T/M: Same meaig as i Table 1. I: Number of iteratios i the phase or the whole algorithm. 5. Coclusios ad future research This study addressed the problem of retrievig cotaiers from the cotaier yard. A three-phase hybrid algorithm was proposed to solve the problem, which aims to miimize the umber of cotaier movemets, as orig time. The algorithm starts by geeratig a iitial feasible retrievig sequece accordig to heuristic rules. Phase two attempts to fid more alterative movemets by six ideas. I the third phase, the algorithm uses dyamic programmig based o a etwor to reduce the total worig time the crae eeds to complete the etire sequece without icreasig the umber of movemets. The two later phases are both iterative, ad termiate whe a umber of cosecutive iteratios caot further improve the curret solutio. Numerical results show that the algorithm is able to solve istaces of more tha 2000 cotaiers, ad thus of practical use to the idustry. Besides, the umber of movemets i the optimal solutios are close to their lower bouds. I this paper, we ivestigated the cotaier retrievig problem of oe RMGC with a sigle spreader. However, craes with multi-spreader are becomig icreasigly popular i termials. Moreover, there are more tha two

12 Zha Bia ad Zhi-hog Ji / Procedia - Social ad Behavioral Scieces 96 ( 2013 ) craes mouted o the same set of rails to wor together for oe tas. The extesio to the case of multiple types of RMGC is a promisig topic for future research. Acowledgemets This research was partially fuded by Natioal Natural Sciece Foudatio of Chia (No : Research o Sychroous Resources Schedulig i a Cotaier Termial based o Hybrid Flow Shop Arragemet), ad by Dalia Sciece ad techology project (No.2012A17GX125: Research o critical techologies for reduced cotaiers relocatio i a cotaier termial). Refereces Lee Y-S, Hsu N-Y. (2007). A optimizatio model for the cotaier pre-marshalig problem. Computers ad Operatios Research, 34(11): Lee Y-S, Chao S-L. (2009). A eighborhood search heuristic for pre-marshalig export cotaiers. Europea Joural of Operatioal Research, 196(2): Chug Y G, Radhawa S U, McDowell E D. (1988). A simulatio aalysis for a trastaier-based cotaier hadlig facility. Computers & Idustrial Egieerig, 14(2): Kim K Y, Kim K H. (1997). A routig algorithm for a trasfer crae to load export cotaiers oto a cotaiership. Computers ad Idustry Egieerig, (33): Kim K H, Kim K Y. (1999). A optimal routig algorithm for a trasfer crae i port cotaier termials. Trasportatio Sciece, 33(1): Kim K H, Hog G-P. (2006). A heuristic rule for relocatig blocs. Computers ad Operatios Research, 33(4): Lee Y-S, Lee Y-J. (2010). A heuristic for retrievig cotaiers from a yard. Computers ad Operatios Research, 37: