SCHEDULING FOR YARD CRANES BASED ON TWO-STAGE HYBRID DYNAMIC PROGRAMMING

TRASPORT ISS 648-442 / eiss 648-3480 208 Volume 33(2): 408 47 doi:0.3846/648442.206.255993 SCHEDULIG FOR YARD CRAES BASED O TWO-STAGE HYBRID DYAMIC PROGRAMMIG Zhan Bian, Qi Xu 2, a Li 3, Zhihong Jin 4 College of Business Administation, Capital Univesity of Economics and Business, China 2, 3, 4 College of Tanspotation Management, Dalian Maitime Univesity, China Submitted 8 Septembe 203; esubmitted 4 ovembe 203, 7 Febuay 204, 4 Decembe 204; accepted Januay 205; published online 08 Decembe 206 Abstact. Making opeational plans fo Yad Canes (YCs) to enhance pot efficiency has become vital issues fo the containe teminals. This pape discusses the load-scheduling poblem of multiple YCs. The poblem is to schedule two YCs at diffeent containe blocks, which seve the loading opeations of one quay cane so as to minimize the total distance of visiting paths and the make-span at stack aea. We conside the containe handling time, the YC visiting time, and the waiting time of each YC when evaluating the make-span of the loading opeation by YCs. Both the containe bay visiting sequences and the numbe of containes picked up at each visit of the two YCs ae detemined simultaneously. A mathematical model, which consides intefeence between adjacent YCs, is povided by means of time-space netwok to fomulate the poblem and a two-stage hybid algoithm composed of geedy algoithm and dynamic pogamming is developed to solve the poposed model. umeical expeiments wee conducted to compae pefomances of the algoithm in this study with actual scheduling ules. Keywods: containe teminals; load-scheduling; geedy algoithm; dynamic pogamming; two-stage hybid algoithm. Intoduction As a hub fo containe tanspotation, containe teminal is an impotant logistics node, linking containe tanspotation by wate and on land. Pot opeations can be geneally divided into two pats: the dischaging opeation duing which containes ae unloaded fom containeships, and the loading opeation duing which containes ae loaded onto ships. In ode to impove sevice levels and economic benefits, containe teminal has to make full use of vaious dischaging and unloading teminal esouces to conduct a easonable scheduling job in ode to impove the teminal s opeational efficiency. In most containe teminals, a lage potion of the tunaound time of a vessel is consumed by the two pocesses. One of the most effective methods fo educing the teminal tunaound time is to impove the poductivity of handing activities. Thee ae thee main types of equipment involved in the dischaging and loading pocess, i.e., Quay Cane (QC), Yad Tuck (YT) and Yad Cane (YC). The planning pocess of containeship opeations consists of a beth scheduling, QC scheduling, and dischage/load sequencing. Though the pocess of beth scheduling, the bething time and position of a containeship ae detemined. Duing the QC scheduling pocess, the numbe of QCs to be assigned to the ship and the job sequence of each QC which indicates the ode of the containes needed to be dischaged/ loaded fom/to the ship by it ae specified. Fo the dischaging pocess, inbound containes ae stacked one by one at a designated empty yad space without consideing the attibutes of each containe. Hence, the YCs do not need to move much and the tunaound time mainly depends on the numbe of inbound containes. Howeve, since the outbound containes ae usually scatteed in the containe blocks and the containes picked up by YCs need to satisfy the job sequences of the QCs, the loading pocess becomes cucial in detemining the tunaound time. As the main handling equipment of containe teminals, YCs play a vital ole in teminal opeation system and YCs oute scheduling diectly impacts the Coesponding autho: Zhan Bian E-mail: bianzhan990@63.com Copyight 206 Vilnius Gediminas Technical Univesity (VGTU) Pess http://www.tandfonline.com/tra

409 Z. Bian et al. Scheduling fo yad canes based on two-stage hybid dynamic pogamming dischaging and unloading efficiency and the opeation of othe tanspotation equipment on pot. Meanwhile, YCs oute scheduling is also affected by othe opeational factos in the teminal system, such as the sequence of dischaging and loading opeations of QCs, the stacking location of containes on the yad, etc. Theefoe, how to detemine YCs outes so as to impove the utilization ate has become a hot topic in the study field of containe teminals. Seveal eseaches have addessed the YC scheduling poblem. It was fist poposed by Chung et al. (988), then Kim and Kim (997, 999) fomulated a mixed intege pogamming model fo the outing poblem of a single-yc loading expot containes out of the stack aea onto waiting YTs. Kim and Kim (2003) applied heuistic algoithms (genetic algoithm and beam seach algoithm) to deal with the same poblem. Thei numeical expeiments showed that the poposed beam seach algoithm outpefoms the genetic algoithm. aasimhan and Paleka (2002) poved that the single-yc outing poblem is P-complete. A heuistic algoithm and an exact banch-bound algoithm fo the poblem wee developed and tested by numeical expeiments. The computational esults showed that the heuistic algoithm is pactical fo the lage-sized poblems. Kim et al. (2003) studied the YC scheduling poblem in anothe way by investigating the sequencing poblem of the YTs. A simulation study was pefomed to compae the pefomance of the poposed solution methods. Consideing the YC opeation constaints, Kim and Pak (2004) employed a banch-bound algoithm fo YC scheduling poblem and developed a geedy andomized adaptive seach pocedue to tackle compute solutions. g and Mak (2005a, 2005b) applied a banch-bound algoithm and heuistic algoithms to minimize the YC waiting time. Han (2005) discussed the optimal oute of a YC while loading containes. He et al. (2007) poposed an intege pogamming and hill climbing algoithm to make the YC scheduling and stoage space allocation an entie system. Howeve, the studies above only focus on the single YC scheduling poblem in which only one YC seves one QC. In fact, due to the pusuit fo highe pot opeation efficiency, it is the common pactice in many containe teminals that two o even moe YCs ae deployed to seve one QC. Theefoe, the scheduling synchonization among YCs becomes moe significant. Zhang et al. (2002) addessed the deployment poblem of YCs, in which yad block that each YC should be assigned to at each time peiod was detemined. Howeve, intefeence among YCs was not consideed in this study, which indicates that YC scheduling poblem could not be solved adically. g (2005) studied the scheduling poblem of YCs consideing intefeence and poposed a heuistic algoithm to minimize the make-span. Howeve, the types of tasks to the loading opeation wee not esticted. Lee et al. (2006, 2007) applied the genetic algoithm and simulated annealing algoithm to solve the same poblem. Wei et al. (2007) studied the YC scheduling in the quantitative opeation condition. Han and Ding (2008) discussed poblems of YCs allocation and optimization in loading/unloading pocess. These ecent eseaches geneally solve multi-ycs scheduling poblems unde static conditions, which ae all single-objective optimizations (eithe make-span o tavel oute), and aely conside the wok schedule of QCs, thus ae not able to visualize scheduling details of YCs. In this pape, a time-space netwok is fomulated to descibe YCs tavel oute, and whee to etieve containes, how many containes to be etieved, which YC to etieve ae consideed in the model. This pape is divided into 6 sections. Following this intoduction, Section defines the YCs loading sequence poblem. A mathematical model is fomulated in Section 2. The famewok and pocedue of twostage hybid optimization algoithm ae developed in Section 3. umeical expeiments ae used to test the pefomance of the poposed method in Section 4. Conclusions ae given in the last section.. YCs Load-Scheduling Poblem Fig. shows a bid-eye s view of a typical containe teminal and it biefly illustates the loading opeation in a containe yad. As illustated in Fig. 2, a block consists of seveal bays whee dozens of containes scatte. A YC is composed of a ganty, a tolley and a speade. When a YC lifts a containe fom the yad, the ganty fist positions itself at the bay whee the taget containe esides, positions its tolley ove the taget stack, lowes its speade to hold onto the containe, and then lifts it up. Afte lifting the containe, the YC can lowe the containe onto a tuck waiting at one end of the cane, o place the containe on the top of anothe stack. In ode to detemine the loading sequence, two documents ae necessay. One is the wok schedule fo a QC (shown in Table ), which shows the sequence of containe goups and the numbe of containes a QC should pick up. A containe goup is defined as a collection of containes of the same length that will be loaded onto the same ship at the same destination pot. The wok schedule fo QC A in Table includes loading tasks to pick up sequentially 36 containes of goup A, 23 containes of goup C, 23 containes of goup A, 24 containes of goup B, 36 containes of goup C, and 36 containes of goup B. The othe document is the yad stowage map, which shows the distibution of containes of each containe goup in the yad as in Table 2. In the poblem, the wok schedule of the QC and the containe yad stowage map ae known befoehand. YC and YC 2 ae employed to seve QC A at the same time. They will pefom the loading opeation accoding to the wok schedule of QC A togethe. It is obvious that diffeent schedules of YCs will leads to diffeent make-span of the loading pocess. The following discusses how to detemine outes of multiple YCs, which ae assigned to the loading op-

Tanspot, 208, 33(2): 408 47 40 YT QC A QC B Block Block 2 YC YC 2 Block 3 Block 4 Fig.. A layout of a containe-loading teminal Cane Tolley Single speade Z (tie) a bay a stack X (bay) Y(stack) Rail Fig. 2. Illustation of the YC and containes in a block of a yad Table. The wok schedule of QC A Sequence 2 3 4 5 6 Goup A C A B C B Quantity 36 23 23 24 36 36 Table 2. The yad stowage map of expot containes in containe yad Bay numbe 42 45 50 70 72 80 85 Goup B A C B A C B Quantity 30 26 40 5 33 9 5 eations fo one QC. The decision vaiables ae the visiting sequence of bays of each YC and the numbe of containes, which each YC picks up at each bay duing the loading opeation. To deal with the visiting sequence of the two YCs is to find the visiting paths of the two YCs, which can be epesented on double-laye timespace netwoks. Fig. 3 is the sample netwoks of YC and YC2 of the example above on which the chaacte in each node (labelled by a squae) is the decision vaiable epesenting whethe the YC paks at the bay with the exception of a stat node and a teminal node. The figue below the node with backets epesents the numbe of containes etieved at the node. The x and y axes specify the sequence and bay espectively. The line between two nodes epesents the visiting path of the YC. The detailed meaning of each chaacte will be discussed in Section 2. Fo sequence, QC A should load 36 containes of goup A, and thee ae two bays (i.e., bay 45 and bay 72) stacking containes of goup A, so y and y 2 ae geneated. It is emakable that the nodes should be divided when containes of the same goup spead ove seveal bays in ode to avoid loops (e.g. each node of bays 42, 70, 85 seving fo sequence 6 is divided into 2 nodes, shown as y 5 y 20 in Fig. 3). This appoach will help YCs move, pak and etieve containes among diffeent bays until the accumulative quantity matches the numbe of containes of a given sequence. Fo each sequence which is concened, m the nodes of the sequence will be divided n times while the quantity of bays to a specific containe goup and the numbe of YCs ae m and n espectively.

4 Z. Bian et al. Scheduling fo yad canes based on two-stage hybid dynamic pogamming Table. e wok schedule of QC A Table 2. e yad stowage map of expot containes in containe yad Bay numbe 42 45 50 70 72 80 85 Goup B A C B A C B Quantity 30 26 40 5 33 9 5 Bay 42 S 45 50 70 72 80 85 Sequence 2 6 z s Sequence 2 3 4 5 6 Goup A C A B C B Quantity 36 23 23 24 36 36 z s 2 y ( ) z 3 x z y 3 4 x 3 z 23 y 2 x ( ) 2 z 24 y 4 x 4 ( ) ( ) y 5 x ( ) 5 z 58 y 8 x ( ) 8 y 6 y 9 x x ( ) ( ) 6 9 y y 20 7 ( x 7 ) ( x 20 ) z 8T z 9T z 20T T YC visiting path Fig. 3. The time-space netwoks of the visiting paths of two YCs YC 2 visiting path 2. Mathematical Fomulation Since the loading sequences ae distibuted among the two YCs, making thei woking schedule dependent on one anothe, the schedules of the two YCs need to be coodinated to minimize the make-span. The following assumptions ae intoduced fo the definition of the poblem: two YCs seve one QC woking fo a vessel; cossove between YCs is not allowed duing the loading pocess; the minimum space between adjacent YCs is equied fo avoiding the collision (in the pape, it was assumed to be one bay); the time equied fo a YC to load a containe is assumed to be the same fo all the containes despite the exact stoage positions of individual containes; thee is only one goup of containe stacked in one bay, which is the common pactice in allocating space in the stack aea of containe teminals; all the containes fo loading opeations ae located in one block o adjacent blocks in the tavel diection. To fomulate the poblem, the following notations will be used: Paametes: J the set of bays; I j the initial numbe of containes stacked at bay j; P the set of sequences; q p the total numbe of containes loading in sequence p; M a sufficiently lage constant; w n the weight of the objective n; R the set of YCs; L the set of lines in the YC netwok; the set of nodes in the YC netwok; the set of nodes in the YC netwok, exclusive ST of the stat node S and the teminal node T; J j the set of nodes within the limits of bay j in the YC netwok; P p the set of nodes of sequence p in the YC netwok; C pf the set of adjacent nodes of etieving action f in sequence p in the YC netwok; CL the set of cossed lines among the sequences o the divided nodes of one sequence in the YC netwok; c ij the tavel distance fom bay i to bay j in the YC netwok; (i, j) the line between node i and node j; [j, k] node j and node k; n = ; n = ; 0 othewise. S T Decision vaiables: x the numbe of containes etieved at node i by YC ; i y i =, if YC paks at node i; 0 othewise; z ij =, if YC moves fom node i to node j; 0 othewise.

Tanspot, 208, 33(2): 408 47 42 As the loading opeation is assigned to two YCs, it is necessay to make them seve one cetain sequence simultaneously and balance thei wokload to minimize the make-span. Besides, fequent moving of YCs will bing inteupt to the loading pocess and affect woking pefomance. Thus, to educe the paking fequency of YCs is also of geat impotance to educe the timeconsumption. What is moe, etieving pattens vay to satisfy the QC loading demands, which can low makespan with the shotest visiting path. Theefoe, in this pape the thee following objectives ae focused: balancing wokload of two YCs, minimizing the paking fequency duing the loading pocess and minimizing the tavel distance of two YCs. The loading sequence poblem can be fomulated by time-space netwok as follows: Object to: min w 2 xi x i + p P i P 2 p i P p w 2 yi + w 3 cij z ij Ri R( ij, ). () L Subject to: fo zij zji = ni { j: ( i, j ) L} { j: ( ji, ) L} z 2 ij + zij i: 2 ( ij, ) L i: ij, L fo j { } { ( ) } 2 zij + zik fo [ j k] 2 { i: ( i, j) L} { i: ( i, k) L } 2 z + z fo ( i, j),( k, l ) CL i ; (2) ; (3), C pf ; (4) ij kl ; (5) zij My fo i R, i ST ; (6) { j: ( ij, ) L } x i My fo i R, i ST ; (7) x = q ; (8) fo p P i p Ri P p x = I fo j J; (9) j Ri J i j i 0 i { 0,} ij { 0,} x fo R, i ; (0) y fo R, i ; () z fo R, ( i j), L. (2) The objective function Eq. () consists of thee polynomials with diffeent weights which is aimed at balancing the wokload of two YCs, minimizing the total numbe of paking times, and minimizing the total tavel distance espectively. Eq. (2) epesents flow consevation of each time-space netwok. Eqs (3) (4) guaantee that two YCs will not collide duing loading pocess. As the two-laye time-space netwoks ae of the same stuctue, one bay is epesented by two coesponding nodes lying in two layes. Eq. (3) esticts the inflow and uppe limit of evey two-node goup to ensue that each bay is seved only by one YC at a time. Eq. (4) esticts the same content fo each etieving opeation of each sequence to guaantee that two YCs can keep a safe distance. Eq. (5) implies that two YCs move in pope ode and ae not pemitted to tavel though one anothe. Eqs (6) (7) defines the moving, paking and etieving ules of YCs. Eq. (6) denotes that only the bay lying in the visiting path can be paked. Eq. (7) means that only when the YC paks can the containes of the paked bay be etieved. Eq. (8) detemines that only if one sequence is completed can the next sequence begin. Eq. (9) is the numbe constaint fo stocked containes, i.e., fo each bay, the total amount of containes etieved fom it is equal to the initial numbe of containes stacked in the bay when all sequences ae completely finished. Eqs (0) (2) ae the non-negative intege constaint of the decision vaiables. 3. Two-Stage Hybid Algoithm As it is well known, the single YC loading scheduling poblem is an P-complete poblem. eedless to say, the poblem pesented in this pape is also an P-complete poblem which makes simple exact algoithms lacking of pacticability to deal with lage-sized cases. Theefoe, hybid algoithms ae equied to solve the poblem efficiently. Bian et al. (206) poposed a hybid optimization algoithm based on dynamic pogamming to solve the loading sequence poblem. It can be divided into two phases, namely, the tavese phase and dynamic pogamming phase. At the tavese phase, a tavese algoithm based on heuistic ules is developed to etieve containe subsets, which need no elocation diectly onto the ship; at the dynamic pogamming phase, a dynamic pogamming with heuistic ules is applied to solve the loading sequence poblem fo the est of the containes. Duing the second phase, heuistic ules ae used to educe the complexity. In this pape, we poposed a new algoithm defined as the two-stage hybid algoithm. The algoithm consists of two phases, namely, the geedy algoithm and the dynamic pogamming. The fome is used to obtain feasible solutions and the latte fo the best solution. Besides, the absolute value function poposed in Section 2 is difficult to be dealt with. By solving the poblem, although we did not change the fom of the objective function, we added some heuistic ules when pogamming the poposed algoithm, i.e. keeping the two YCs woking simultaneously as fa as possible to maintain thei wokload balanced. 3.. Definition of ode In ode to descibe the poposed algoithm pecisely, a new concept node is intoduced in this pape, which epesents the geneal states of the yad and the YCs at a time. Moe specifically, a node impots two types of infomation of which one is the yad infomation (i.e. the total numbe of emaining containes in the yad, the numbe and the categoy of emaining containes in each bay), the othe is the YCs infomation (i.e. the

43 Z. Bian et al. Scheduling fo yad canes based on two-stage hybid dynamic pogamming loading sequence, cuent position, total tavel distance, etieving quantity and paking times of the two YCs). A new node J will be geneated afte a node I going though a seial of opeations, and I, J can be defined as a fathe node and a child node espectively. It is obvious that the numbe of emaining containes of a child node J is no moe than that of a fathe node I. The poposed hybid algoithm which is aimed at educing the amount of emaining containes is divided into two stages. In the fist stage, geedy algoithm is poposed to geneate an initial feasible solution; in the second stage, dynamic pogamming is suggested to find the optimal solution. In othe wods, accoding to the scheduling ules, the fathe node is divided into seveal child nodes (i.e. one o moe) which will be pocessed to get a solution. The solution coesponding with the child node of the least objective function value (heeafte called cost) is the initial feasible solution while all child nodes should be consideed in ode to get the optimal solution. As peviously mentioned, a feasible solution is obtained when thee is no containe left. Then the cost of the feasible solution will be calculated, and a feasible solution tuns into an optimal solution if its cost is no moe than that of othe feasible solutions. The final pupose of the two-stage hybid algoithm is to find an optimal solution. 3.2. Geedy Algoithm In this section, a feasible solution is geneated by the following steps: Step 0: build a node queue and join initial nodes to the tail of the queue; Step : set the head node as I, and go to Step 2; Step 2: if thee is no containe to be loaded in I, then set I as the feasible solution, and the algoithm is teminated; othewise, go to Step 3. Step 3: accoding to scheduling ules, I is divided into seveal Child odes, ecoded as J and so on; if thee exists seveal child nodes, then sot the nodes in an ascending ode based on diffeent costs and go to Step 4. Step 4: join the child node with the least cost to the tail of the node queue, and go to Step. Fig. 4 illustates the pocess to obtain an initial feasible solution. 3.3. Dynamic Pogamming In dynamic pogamming, the pocess of geneating child nodes can be egaded as the pocess in which one -dimensional poblem is conveted into k( ) -demensional sub-poblems, and the figue k is unpedictable. Thus, a seial of new child nodes can be obtained. The quantity of emaining containes deceases till no containe is left. Duing the pocess, the dynamic equation can be fomulated as follows: cost (I) = min (cost (I ) + cost (K)_, cost (I 2 ) + cost (K)_2, ), (3) whee: I epesents the fathe node; J, J 2, ae child nodes geneated by I afte scheduling opeations; K indicates the numbe of containes involved in the scheduling pocess (i.e. the fathe node contains K moe containes than the child nodes); cost K epesents the cost as a esult of convesion fom the fathe node to the child nodes. Duing each stage of dynamic pogamming, seveal new states ae geneated. Thus, the numbe of states duing computing pocess will explosively incease which will lead to a lage amount of opeations. Consideing the complexity of the algoithm, a lopping opeation to deal with the nodes is suggested, i.e., ignoing the child node whose cost is geate than the existing lowest cost in each stage. Fig. 5 shows the famewok of dynamic pogamming to obtain an optimal solution. To be moe pecise, the algoithm can be descibed as follows: Step 0: set the cost of the initial feasible solution as cost. Build a ode queue and join initial odes to the tail of the queue; Step : if the queue is empty, then the algoithm is teminated; othewise, go to Step 2; Step 2: set the head ode as I, and set the cost of I as cost and go to Step 3; Geneate Js, sot in an ascending ode Join J with the least cost to the tail of the queue Build a ode queue, join initial odes to the tail of the queue Set the head ode as I o containe le in I Set I as the feasible solution, the algoithm is teminated Fig. 4. The famewok of geedy algoithm Y Empty S, join Iў to S, cost costў costў< cost Y Y Fig. 5. The famewok of dynamic pogamming Y Build a ode queue, join initial odes to the tail of the queue e queue is empty Set the head ode as Iў and set its cost as costў costў > cost Iў is the Final ode Join Iў to S Y e ode in S is the optimal solution and the algoithm is teminated Iў divides into seveal Child odes

Tanspot, 208, 33(2): 408 47 44 Step 3: compae cost with cost, if the fome is lage than the latte, which means the optimal solution will not be obtained afte I, then ignoe I and go to Step ; othewise, go to Step 4; Step 4: if cost equals cost and I is a final node, then join I to the set of optimal solutions S and go to Step ; othewise, go to Step 5; Step 5: if cost is smalle than cost and I is a final node, then empty S and join I to the set and eplace cost by cost and go to Step ; othewise, go to Step 6; Step 6: if cost is smalle than cost and I is no longe a final node, then I is divided into seveal child nodes accoding to scheduling ules and join all the child nodes to the tail of the queue and go to Step. 4. umeical Expeiments 4.. Expeiment Design In this section, computational examples to demonstate the pefomance of the algoithm developed in this pape ae povided. The poposed algoithm has been implemented in Micosoft Visual C++ and un on a pesonal compute, which has an InteCoe I5 CPU (320M) unning at 2.50GHz and with 4.0GB memoy RAM. Sample poblems ae geneated based on the following peconditions: the numbe of containes fo each sample anges fom 00 to 500; fo each sample poblem, the containes ae andomly classified into seveal goups, namely A, B, C, and so foth; the wok schedule of the QC is geneated by joining these containe goups in a andom sequence; containes equied by the QC ae allocated andomly in one o moe blocks, each of which consists of 35 bays subjected to the constaint that only one goup of containe can be stacked in a bay. umeical expeiments ae oganized by two sections: the fist section is an actual case analysed with detailed esults; the second section begins with a numbe of small-sized expeiments followed by diffeent lagesized tests designed to validate the effectiveness of the algoithm. Duing actual opeations of containe teminal, YCs ae always andomly scheduled, i.e., two YCs opeate andomly unde the pemise of safety constaints without the consideation of balancing the wokload and minimizing paking fequency. In this pape, such usual pactice is defined as actual ules and also pogammed to get actual solutions set as contolled tials. Some pefeence settings ae shown as follows: the tavel distance between two adjacent bays is 7 metes, the distance between two YCs emains at least 2 metes fo safety pupose at anytime, the moving speed of YCs is 5 metes pe second, and the time equied fo a YC to load a containe is 2 minutes. In ode to conduct tial compaisons expediently, w, w 2, w 3 ae set to 0.4, 0.4, 0.2 espectively. 4.2. An Actual Case Actual data fom DCT (Dalian Containe Teminal, China) wee collected and pocessed to build a case as shown in Tables and 2. Assuming that YC and YC2 pak at bay 45 and bay 72 espectively when the loading opeation begins. The following shows the computational esults solved by actual ules and the poposed algoithm espectively. Table 3 illustates the solution fo two-yc scheduling poblem accoding to actual ules, while Table 4 epesents the solution by the two-stage hybid algoithm. It is obvious that the make-span via the poposed algoithm is shote than that of actual ules (206.653 vs. 226.770 mins). The pefomance of wokload balance via the poposed algoithm is much bette than that of actual ules (4 vs. 42 imbalances). The paking fequency deceases distinctly fom 9 to times. Besides, the total tavel distance and of couse the objective function value both see a damatic decline compaing actual ules with two-stage hybid algoithm. In geneal, by compaing figues of Table 3 with that of Table 4, optimization levels obtained by the poposed algoithm of the wokload balance, paking fequency, tavel distance, and make-span ae 66.7, 42., 3.5, 9.7% espectively. Table 3. The scheduling esults of YCs by actual ules YC o Wok time [min] Wokload [TEU] Paking fequency Tavel distance [m] Objective function value (cost) 220.607 0 8 224 2 226.770 68 399 Total 226.770 78 (sum)/ 42 (diffeence) 9 623 49 Table 4. The scheduling esults of YCs by two-stage hybid algoithm YC o Wok time [min] Wokload [TEU] Paking fequency Tavel distance [m] Objective function value (cost) 206.49 96 5 203 2 206.6533 82 6 336 Total 206.6533 78(sum)/ 4 (diffeence) 539 7.8

45 Z. Bian et al. Scheduling fo yad canes based on two-stage hybid dynamic pogamming Moeove, the poposed two-stage hybid algoithm allows the scheduling pocess to be visual to the wokes by outputting the details of the solution, as shown in Tab le 5. The time-space infomation can also be obtained duing the computational pocess, which is illustated in Table 6. The column space between YCs explains that two YCs ae unde safety constaints duing loading opeation. 4.3. Vaious-Sized Expeimental Tests In this section, two types of cetain-sized poblems wee developed featuing the amount of bays (i.e. small-sized: less than 0 bays, lage-sized: to 20 bays). Fo each poblem of a given numbe of bays, 20 vaious expeiments wee conducted to obtain an aveage value of the indexes, of which tavel distance and make-span ae much moe concened duing loading opeations. Thus, the compaisons of these two indexes between the poposed two-stage hybid algoithm and the actual ules ae specially epesented as follows. Figs 6 and 7 show that fo small-sized poblems, diffeences of the tavel distance and the make-span between two algoithms ae both conspicuous. On aveage, the tavel distance and the make-span obtained fom the two-stage hybid algoithm ae 3.87, 0.22% shote than that of the actual ules espectively. Table 5. The scheduling details of two YCs The scheduling details of YC Sequence Stat time [min] End time [min] Time consuming [min] Action* Pak (Y/) Tavel distance [m] 0 0.000 36.000 36.000 R: 45, 8, A 0 36.000 36.7 0.7 M: 45, 50 Y 35 2 36.7 60.7 24.000 R: 50, 2, C 0 3 60.7 60.234 0.7 M: 50, 45 Y 35 4 60.233 76.233 6.000 R: 45, 8, A 0 5 76.233 78.6.883 W: 45,.883 0 6 78.7 78.87 0.070 M: 45, 42 Y 2 7 78.87 90.7.930 W: 42,.930 0 8 90.7 4.7 24.000 R: 42, 2, B 0 9 4.7 4.304 0.87 M: 42, 50 Y 56 0 4.303 70.303 56.000 R: 50, 28, C 0 70.303 70.490 0.87 M: 50, 42 Y 56 2 70.490 206.490 36.000 R: 42, 8, B 0 Total 206.490 5 203 The scheduling details of YC 2 Sequence Stat time [min] End time [min] Time consuming [min] Action Pak (Y/) Tavel distance [m] 0 0.000 36.000 36.000 R: 72, 8, A 0 36.000 36.87 0.87 M: 72, 80 Y 56 2 36.87 58.87 22.000 R: 80,, C 0 3 58.87 58.373 0.87 M: 80, 72 Y 56 4 58.373 60.7.744 W: 72,.743 0 5 60.7 90.7 30.000 R: 72, 5, A 0 6 90.7 90.63 0.046 M: 72, 70 Y 4 7 90.63 4.63 24.000 R: 70, 2, B 0 8 4.63 4.397 0.234 M: 70, 80 Y 70 9 4.397 30.397 6.000 R: 80, 8, C 0 0 30.397 32.303.906 W: 80,.907 0 32.303 32.420 0.7 M: 80, 85 Y 35 2 32.420 70.303 37.883 W: 85, 37.883 0 3 70.303 200.303 30.000 R: 85, 5, B 0 4 200.303 200.653 0.350 M: 85, 70 Y 05 5 200.653 206.653 6.000 R: 70, 3, B 0 Total 206.653 6 336 otes: in column Action, R: 45, 8, A epesents Retieve 8 containes of goup A at bay 45 ; M: 45, 50 epesents Move fom bay 45 to bay 50 ; W: 45,.883 epesents Stay at bay 45 fo.883 min.

Tanspot, 208, 33(2): 408 47 46 Table 6. The time-space details of two YCs Time [min] Space of YC (bay) Space of YC 2 (bay) Space between YCs (counted by bay) 0.000 45 72 27 36.000 45 72 27 36.7 50 77 27 36.87 50 80 30 58.87 50 80 30 58.373 50 72 22 60.7 50 72 22 60.233 45 72 27 76.233 45 72 27 78.7 45 72 27 78.87 42 72 30 90.7 42 72 30 90.63 42 70 28 4.7 42 70 28 4.63 44 70 26 4.303 50 76 26 4.397 50 80 30 30.397 50 80 30 32.303 50 80 30 32.420 50 85 35 70.303 50 85 35 70.490 42 85 43 200.303 42 85 43 200.653 42 70 28 206.490 42 70 28 Similaly, the two-stage hybid algoithm is able to achieve bette solutions fo lage-sized poblems compaed with the actual ules, as shown in Figs 8 and 9. The esults of tavel distance and the make-span ae 8.2, 8.0% bette than the actual ules espectively. In conclusion, the two-stage hybid algoithm is appopiate to the stuctue of the solution space of the load-scheduling poblem in this pape and a moe pomising solution method compaed with the actual ules. 400 350 300 250 200 50 00 50 0 The amount of bays Two-stage hybid algoithm Tavel distance [metes] Actual ules 5 6 7 8 9 0 20 44 90 220 245 335 57 62 208 235 35 379 Fig. 6. Compaison of tavel distance between two-stage hybid algoithm and actual ules fo small-sized cases The make-span [min] 0 The amount of bays Two-stage hybid algoithm Actual ules 300 250 200 50 00 50 5 6 7 8 9 0 85 8 32 64 96 244 00 20 54 76 220 276 Fig. 7. Compaison of make-span between two-stage hybid algoithm and actual ules fo small-sized cases Tavel distance [m] 0 The amount of bays Two-stage hybid algoithm Actual ules 4000 3500 3000 2500 2000 500 000 500 632 732 2 3 4 5 6 7 8 9 20 798 9 284 63 200 2340 2804 300 3476 903 02 49 934 2646 2700 2990 3420 3785 Fig. 8. Compaison of tavel distance between two-stage hybid algoithm and actual ules fo lage-sized cases The make-span [min] 0 The amount of bays Two-stage hybid algoithm Actual ules 600 500 400 300 200 00 264 299 2 3 4 5 6 7 8 9 20 298 3 325 34 39 40 455 479 523 335 364 370 386 4 435 47 50 54 Fig. 9. Compaison of make-span between two-stage hybid algoithm and actual ules fo lage-sized cases Conclusions A load-scheduling poblem of expot containes was intoduced fo the case that multiple YCs and one QC ae used in containe teminals. The load-scheduling poblem simultaneously detemines thee key factos: whee to pick up containes, how many containes should be picked up and how the YC moves among bays duing the loading opeation. It was attempted to minimize the make-span of loading tasks by the means of balancing wokload of YCs, educing the paking fequency, and minimizing the tavel distance. A mathematical model was developed fo the load-scheduling poblem. In ode to evaluate the objective function, the containe handling time at a bay, the tavel distance between two

47 Z. Bian et al. Scheduling fo yad canes based on two-stage hybid dynamic pogamming adjacent bays and the waiting time due to intefeence among YCs wee consideed. A two-stage hybid algoithm fo optimal solutions was poposed to solve the poblem. Usual opeations obtained by actual ules wee conducted as contolled tials. Both small and lage-sized expeiments showed that the two-stage hybid algoithm outpefoms the actual ules in thei aveage make-span and aveage objective function value. The load sequence of individual containes within a specific bay has not been teated in this pape, which is a topic fo ou futhe eseach. Acknowledgments This wok was patially suppoted by ational atual Science Foundation of China (o 77208, o 7302044), Doctoal Pogam Foundation of Institutions of Highe Education of China (o 2022250009), the Fundamental Reseach Funds fo the Cental Univesities (o 33203320, o 33203076) and Postdoctoal Science Foundation of China (o 203M530927). Refeences Bian, Z.; Shao, Q.; Jin, Z. 206. Optimization on the containe loading sequence based on hybid dynamic pogamming, Tanspot 3(4): 440 449. http://doi.og/0.3846/648442.204.994563 Chung, Y.-G.; Randhawa, S. U.; McDowell, E. D. 988. A simulation analysis fo a tanstaine-based containe handling facility, Computes & Industial Engineeing 4(2): 3 25. http://doi.og/0.06/0360-8352(88)90020-4 Han, X. L. 2005. Routing poblem of tansfe cane at containe teminals, Jounal of Shanghai Maitime Univesity 26(2): 39 4. (in Chinese). Han, X.-L.; Ding, Y.-Z. 2008. Yad cane allocation and optimization in containe teminal, avigation of China 3(): 6 4. (in Chinese). He, J.; Mi, W.; Yan, W. 2007. Containe yad cane scheduling based on hill-climbing algoithm, Jounal of Shanghai Maitime Univesity 28(4): 5. (in Chinese). Kim, K. H.; Kim, K. Y. 999. An optimal outing algoithm fo a tansfe cane in pot containe teminals, Tanspotation Science 33(): 7 33. http://doi.og/0.287/tsc.33..7 Kim, K. Y.; Kim, K. H. 2003. Heuistic algoithms fo outing yad-side equipment fo minimizing loading times in containe teminals, aval Reseach Logistics 50(5): 498 54. http://doi.og/0.002/nav.0076 Kim, K. Y.; Kim, K. H. 997. A outing algoithm fo a single tansfe cane to load expot containes onto a containeship, Computes & Industial Engineeing 33(3 4): 673-676. http://doi.og/0.06/s0360-8352(97)0029-2 Kim, K. H.; Lee, K. M.; Hwang, H. 2003. Sequencing delivey and eceiving opeations fo yad canes in pot containe teminals, Intenational Jounal of Poduction Economics 84(3): 283 292. http://doi.og/0.06/s0925-5273(02)00466-8 Kim K. H.; Pak, Y. M. 2004. A cane scheduling method fo pot containe teminals, Euopean Jounal of Opeational Reseach 56: 752 768. http://doi.og/0.06/s0377-227(03)0033-4 Lee, D.-H.; Cao, Z.; Meng, Q. 2007. Scheduling of two-tanstaine systems fo loading outbound containes in pot containe teminals with simulated annealing algoithm, Intenational Jounal of Poduction Economics 07(): 5 24. http://doi.og/0.06/j.ijpe.2006.08.003 Lee, D.-H.; Meng, Q.; Cao, Z. 2006. Scheduling two-tanstaine systems fo loading opeation of containes using evised genetic algoithm, in TRB 85th Annual Meeting Compendium of Papes CD-ROM, 22 26 Januay 2006, Washington, DC, US, 3. aasimhan, A.; Paleka, U. S. 2002. Analysis and algoithms fo the tanstaine outing poblem in containe pot opeations, Tanspotation Science 36(): 63 78. http://doi.og/0.287/tsc.36..63.576 g, W. C. 2005. Cane scheduling in containe yads with inte-cane intefeence, Euopean Jounal of Opeational Reseach 64(): 64 78. http://doi.og/0.06/j.ejo.2003..025 g, W. C.; Mak, K. L. 2005a. An effective heuistic fo scheduling a yad cane to handle jobs with diffeent eady times, Engineeing Optimization 37(8): 867 877. http://doi.og/0.080/0305250500323849 g, W. C.; Mak, K. L. 2005b. Yad cane scheduling in pot containe teminals, Applied Mathematical Modelling 29(3): 263 267. http://doi.og/0.06/j.apm.2004.09.009 Wie, Z.; Shen, J.; Xiao, R.; Zhang, Z.; Shi, D. 2007. Reseach on ubbe tied ganty cane scheduling of pot containe teminal, Engineeing Sciences 9(8): 47 5. (in Chinese). Zhang, C.; Wan, Y. W.; Liu, J.; Linn, R. J. 2002. Dynamic cane deployment in containe stoage yads, Tanspotation Reseach Pat B: Methodological 36(6): 537 555. http://doi.og/0.06/s09-265(0)0007-0