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Dynamic crane deployment in container storage yardsChuqian Zhang a ,Yat-wah Wan a ,Jiyin Liua,*,Richard J.Linn b aDepartment of Industrial Engineering and Engineering Management,The Hong Kong University of Science and Technology,Clear Water Bay,Kowloon,Hong Kong b Department of Industrial and Systems Engineering,Florida International University,Miami,FL,USAReceived 7February 2000;received in revised form 2April 2001;accepted 9April 2001AbstractStorage yards at container terminals serve as temporary buffers for inbound and outbound containers.Rubber tyred gantry cranes (RTGCs)are the most frequently used equipment in yards for container han-dling.The efficiency of yard operations heavily depends on the productivity of these RTGCs.As the workload distribution in the yard changes over time,dynamic deployment of RTGCs among storage blocks (container stacking areas)is an important issue of terminal operations management.This paper addresses the crane deployment problem.Given the forecasted workload of each block in each period of a day,the objective is to find the times and routes of crane movements among blocks so that the total delayed workload in the yard is minimized.The problem is formulated as a mixed integer programming (MIP)model and solved by Lagrangean relaxation.To improve the performance of this approach,we augment the Lagrangean relaxation model by adding additional constraints and modify the solution procedure putational experiments show that the modified Lagrangean relaxation method generates excellent solutions in short time.Ó2002Elsevier Science Ltd.All rights reserved.Keywords:Container terminal;Crane deployment;Mixed integer programming;Lagrangean relaxation1.IntroductionA container terminal plays an important role in global manufacturing and international business by serving as a multi-modal interface,usually between the sea and land transports.Its three basic functions are as follows:(a)delivering containers to (carriers for)consignees and receiving containers from shippers,(b)loading containers onto and discharging containers from vessels,and (c)storing containers temporarily to account for the differences in arrival times ofthe Transportation Research Part B 36(2002)537–/locate/trb*Corresponding author.Tel.:+852-2358-7100;fax:+852-2358-0062.E-mail address:jyliu@ust.hk (J.Liu).0191-2615/02/$-see front matter Ó2002Elsevier Science Ltd.All rights reserved.PII:S0191-2615(01)00017-0sea and land carriers.A container terminal works under multiple operational objectives.Two main ones are to minimize the ship turnaround time and to maximize the terminal throughput.They can be achieved by efficient loading and discharging of vessels.Therefore,quay crane (QC)rate is usually taken as the key performance measure for the operational efficiency of a terminal.Let Q be the total number of QCs used in period [0,t ];S i and C i be the start time and the completion time of the i th QC,respectively;W be the total number of containers loaded and discharged in [0,t ].Then the QC rate in [0,t ]is W =P Q i ¼1ðC i ÀS i Þ.The operations of QCs,and hence the QC rate,are the results of several interrelated container flows in a container terminal.A schematic diagram of these flows is shown in Fig.1in which we assume a typical sea-land inter-modal interface.Without loss of generality,we take trucks as the means of land transport.Other means such as trains have similar processes.An inbound container is brought in by a vessel and is to be loaded on a truck,while an outbound container is brought in by a truck and is to be loaded onto a vessel.Both inbound and outbound flows take three handling stages in a terminal,namely the gatehouse ,the storage yard ,and the quayside .On the land side,external trucks (XTs)bring in outbound containers before the loading process of the relevant vessel,and they pick up inbound containers after the discharging process of the vessel.All XTs first go through the gatehouse for documentation and,if applicable,con-tainer inspection.After dropping offthe outbound containers and/or picking up the inbound containers in the yard,they leave the terminal through the gatehouse.On the quayside,vessels bring inbound containers to the terminal and leave with outbound containers on board according to vessel schedules.Each arriving vessel first docks at an assigned berth.Then it is discharged and loaded by QCs.The transportation of discharged inbound containers from the quayside to the storage yard and the transportation of outbound containers from the storage yard to the quayside for loading are carried out by internal trucks (ITs).At the storage blocks in the yard,yard cranes carry out all the container handling work which includes grounding (lifting containers from the trucks,ITs and XTs,and storing them at storage locations)and picking-up (retrieving containers from their storage locations and putting them on the trucks).Because yard cranes serve both ITs and XTs,they play a very important role in the storage yard and are usually the bottleneck in the container handling process.Without an effective operation planning of yard cranes,ITs may have to wait in the yard,and consequently QCs will be idle in waiting for ITs.Such waiting times reduce the QC rate.Therefore,to achieve high productivity,the use of yard cranes should be well planned.There are several kinds of common yard cranes.A rail-mounted gantry crane runs on rails with a single storage block between the rails.A rubber tyred gantry crane (RTGC)runs onrubber-tyre538 C.Zhang et al./Transportation Research Part B36(2002)537–555wheels and can move among storage blocks.See Fig.2in Section 2for an RTGC and a typical storage block.Other cranes such as straddle carriers which carry containers between their legs,mobile harbor cranes which are inherited from pre-containerized era,and heavy-duty forklift trucks which are more used for empty containers are used for different spatial-economical reasons.RTGCs are the most popular type of yard cranes at the container terminals in Hong Kong as well as in many other ports in the world.In this paper we focus on the deployment of RTGCs among blocks to minimize the total work delayed so as to raise the container handling efficiency.The rest of the paper is organized as follows:Section 2describes the dynamic crane deployment problem in detail.A literature review on previous work in the area of container terminal operation management is provided in Section 3.Section 4presents a mixed integer programming (MIP)model of the problem.In Section 5,the Lagrangean relaxation method is used to solve the model.Improvement measures are introduced to the standard procedure to reduce the large duality gap.Section 6reports the results of computational experiments on a large set of representative problem instances.Conclusions are drawn in Section 7.2.Problem descriptionThe container storage yard is used to temporarily store the inbound and outbound containers until they are picked up by trucks or loaded onto vessels.A large-scale yard may be divided into a number of large areas called zones .In each zone,containers are stacked side by side and one on top of the other to form rectangular shaped heaps called blocks .A typical block,as shown in Fig.2,may have 6lanes of containers placed side by side,5containers in height for each lane and usually more than 20containers in length,depending on the geographical shape of the storage yard.Containers’retrieval and stacking within each block are performed by RTGCs.Depending on the vessel schedule and the policy for container allocation in the yard,the workloads of blocks vary among blocks and with time.It is important that the workload can be finished within the planned time periods so that vessels can leave at their scheduled departure times.Since RTGCs are very expensive equipment,terminals usually cannot allocate and keepa C.Zhang et al./Transportation Research Part B36(2002)537–555539fixed number of RTGCs in each block.To fully utilize the RTGCs and overcome the workload imbalance among blocks,an RTGC needs to move to a block with heavy workload after finishing the light workload in its current block.The allocation and the movement of RTGCs among blocks are collectively called crane deployment .As shown in Fig.3,if two blocks are adjacent and align longitudinally,e.g.blocks B 1and B 2,an RTGC can move easily from one block to the other along a straight line without any turning.In other cases,e.g.,to move between blocks B 1and B 3,or between blocks B 1and B 4,an RTGC has to make a 90°turn (of its wheels)twice to move from one block to another.Since RTGCs are big in size and slow in motion,their movements occupy a large amount of road space for a long time,obstructing the traffic on the roads between blocks and delaying other terminal operations.Moreover,any block-to-block RTGC movement takes time,and will results in the loss in the productivity of cranes.Therefore,determining the optimal RTGC deployment among blocks is very critical to the efficiency of container handling operations in the storage yard.The container yard operates three shifts round the clock every day of the year.Each shift is further divided into two 4-h planning periods,leading to a total of six planning periods per day:00:00–4:00,4:00–8:00,8:00–12:00,12:00–16:00,16:00–20:00,and 20:00–24:00.Every evening,the fore-casted workloads in the block for the following day are assembled.The operating plan of RTGCs,including any possible deployment,is determined for each of the six planning periods.Such op-erating and deployment plans are made by crane operation supervisors based on their experience.Basically,given the forecasted workload in each block in each planning period,the supervisors make the following two decisions so as to minimize the total workload overflow between periods:(a)the times to deploy (move)each RTGC,and(b)the routes of RTGC movements (i.e.,the origin and the destination blocks of each RTGC)for each period of the planning horizon.While the supervisors have a good sense of the operations,the size of the deployment problem practically precludes any person from making sound decisions continuously.In our research,we formulate the problem as a mathematical program and provide suggestions for supervisors and terminal management to follow.3.Related workThe QC rate of a container terminal depends on the effective and efficient usage,including allocation and scheduling,of the expensive resources such as berths,QCs,storage space andyard540 C.Zhang et al./Transportation Research Part B36(2002)537–555C.Zhang et al./Transportation Research Part B36(2002)537–555541 cranes.Each of these resources plays an indispensable role in the interlocking processes in a container terminal.See Zhang(2000)for a comprehensive review of literature on various models and aspects of allocating and scheduling of these resources.Here,we only provide a brief review of existing studies related to terminal resources.Berths are essential resources of a terminal.Therefore,under heavy traffic of vessels,it is critical to allocate berths to vessels to avoid undue vessel delay.The berth allocation problem was studied by using non-linear integer programming model(Imai et al.,1997),modified time-space network (Chen and Hsieh,1999)and simulation(Lai and Shih,1992)to minimize the vessel berthing time and maximize berth utilization.QCs are the handling equipment at berths for discharging and loading berthed vessels.Allo-cation and scheduling of QCs directly affect the utilization of berths.Daganzo(1989)used an integer programming model and a simple scheduling strategy to solve the static QC scheduling problem with the objective of minimizing the aggregate vessel delay cost.Peterkofsky and Dag-anzo(1990)considered the problem as an‘‘open shop’’scheduling problem with parallel and identical machines(QCs).An integer programming model was built to minimize ship delay costs (weighted tardiness)and a branch and bound method is used to obtain an optimal solution.Based on the above results,Daganzo(1990)further studied the effect of QC scheduling strategies on terminal throughput and ship delay in the long run,assuming that the workload distribution(over time)for ships at berths is the same as the workload distribution for the ship’s population as a whole.Tabernacle(1995)applied learning concept to QC operations in order to improve the QC performance with time.The yard operation is the most complex part at a terminal because both inbound and out-bound containerflows are handled there simultaneously.Resource planning in the yard deter-mines the terminal efficiency to a great extent.There are two clear research topics related to yard operation:the assignment of storage locations to containers(i.e.,the grounding policy)and the allocation and scheduling of yard equipment.Grounding policy is a very critical issue that ul-timately determines the amount of workload in each storage block.Taleb-Ibrahimi et al.(1993) and De Castilho and Daganzo(1993)compared the amount of required storage space and the resulting workload under different grounding policies for export and import containers respec-tively.Kim and Kim(1994,1996),Kim and Park(1998)and Kim and Kim(1998)developed various mathematical programming and cost models tofind the grounding policy under various criteria.There has not been much research done on crane allocation and i and Lam (1994)used simulation to compare throughputs,equipment utilizations,and waiting times for several allocation strategies of yard equipment.Kim and Kim(1999)minimized the sum of the set up time and the travel time of cranes in a container storage block by modeling the process as a mixed integer program.There is also research that considers several operational decisions simultaneously.Kozan and Preston(1999)determined the optimal grounding policy and the corresponding crane schedule using an integer program to minimize the maximal working time among all yard cranes assigned to a vessel.Ballis and Abacoumkin(1996)built a simulation model that incorporates traffic congestion and delay due to mismatch of equipment.They used the model to evaluate yard layout,equipment amount,truck arrivals,and operations rules of a terminal.Kozan(1997a) studied the throughput of a train container terminal.He developed heuristics to assign trains to542 C.Zhang et al./Transportation Research Part B36(2002)537–555tracks and to assign cranes,forklifts,and highstacks to containers.An analytically based simu-lation was used to evaluate these heuristics for different performance measures.In another paper Kozan(1997b)modeled the operations as a batch-arrival multi-server queue and discussed the major factors affecting the transfer efficiency of container terminals.It is clear from the literature review that there have not been systematic studies on the RTGC deployment problem.Given the popularity of RTGCs in container terminals and their important role in the yard operation,effective RTGC deployment is very important for high terminal effi-ciency.In the following,we will develop an integer programming model to deploy RTGCs so that the total work delayed in the yard is minimized.Near-optimal solutions are then obtained through Lagrangean relaxation heuristic.4.Model development4.1.Modeling assumptionsWe make the following assumptions in our RTGC deployment model.All these assumptions are from observing the real-life practice.•The capacity of an RTGC is measured in crane minutes.All cranes have the same capacity of 240min in each4-h planning period.Similarly,the workload is also measured in crane minutes: Wefirst get the nominal number of container moves,which is the total number of containers to be discharged from and loaded onto vessels,and to be grounded from and picked up by XTs during each planning period.Since containers are stacked on each other and may be stored in a pre-defined pattern,each nominal container retrieval or storage may take more than one real crane move.So the workload,the total number of container moves,is converted into the work-load in crane minutes by multiplying the average number of real moves per nominal move with the average crane time needed per move.•Because of the limitation of block sizes and the potential danger of crane collision,each block holds at most two RTGCs at any time.With at most only two RTGCs per block,we do not allow more than one RTGC moving from one block to another in a period.•In practice,to minimize the loss of crane hours and the traffic blockage,inter-zone RTGC movements are carried out only once per day at midnight.Consequently,in our study of real-time crane deployment in day time,we do not need to consider any inter-zone moves but focus only on intra-zone moves within each24-h planning horizon.•Each RTGC is moved at most once in each4-h planning period.This is the trade offadopted in practice to avoid frequent RTGC moves that block traffic and lose crane hours on one hand and infrequent RTGC moves that waste surplus capacity on the other hand.•Any crane move starts andfinishes within the same planning period.•Unfinished work in a block at the end of a planning period will be carried over to the next pe-riod.As a result,the workload of a block in a planning period is the sum of the workload of the current period and the workload carried over from the previous planning period.The workload carried over from the previous period will befinished during the early part in the current pe-riod.We refer the unfinished workload in a period as‘‘delayed work’’.4.2.Crane deployment modelThe deployment decision process can be formulated as a MIP model to find the optimal movements of cranes and the timing of the movements in the yard (or a zone in a large-scale yard)during each planning period of a day.The following notations are used:X ii 0the number of cranes in block i at the beginning of the planning horizon,Cthe capacity of one crane within a planning period;C ¼240min,Nthe total number of blocks under consideration,Tthe total number of planning periods in a planning horizon;T ¼6periods,B itthe workload in block i within planning period t ,t ij the traveling time of a crane from block i to block j .The decision variables are defined as follows (Fig.4):X ijtthe number of cranes moving from block i to block j during planning period t (note that when i ¼j ,these cranes stay in the same block during period t ),Z ijtthe workload fulfilled in block i by cranes that move from block i to block j during planning period t (refer to Fig.4),Y ijtthe workload fulfilled in block j by cranes that move from block i to block j during planning period t (refer to Fig.4),W it the workload left in block i at the end of planning period t .The model is presented below,followed by a brief explanation.ðCD P ÞMin X T t ¼1X N i ¼1W it :Subject to:XN j ¼1X ijt ¼X N j ¼1X ji ðt À1Þfor i ¼1;2;...;N ;t ¼1;2;...;T ;ð1ÞXN j ¼1X ijt þX N j ¼1j ¼i X jit 62for i ¼1;2;...;N ;t ¼1;2;...;T ;ð2ÞC.Zhang et al./Transportation Research Part B36(2002)537–555543W i ðt À1ÞþB it ÀXN j ¼1Z ijtþX N j ¼1Y jit !ÀW it ¼0for i ¼1;2;...;N ;t ¼1;2;...;T ;ð3ÞZ ijt þY ijt 6ðC –t ij ÞX ijt for i ¼1;2;...;N ;j ¼1;2;...;N ;t ¼1;2;...;T ;ð4ÞW i 0¼0for i ¼1;2;...;N ;ð5ÞX ij 0¼0for i ¼1;2;...;N ;j ¼1;2;...;N ;i ¼j ;ð6ÞW it P 0;Z ijt P 0;Y ijt P 0for i ¼1;2;...;N ;j ¼1;2;...;N ;t ¼1;2;...;T ;ð7ÞX iit 2f 0;1;2g ;X ijt 2f 0;1gfor i ¼1;2;...;N ;j ¼1;2;...;N ;i ¼j ;t ¼1;2;...;T :ð8ÞAs stated in the assumptions,the workload within one planning period can be carried over to the next one if it cannot be finished.However,all jobs in a block have their own due times and ideally should always be completed on time.Although these due times are flexible to some extent,the work should not be delayed for long.Therefore,it is not appropriate to use only the workload left at the end of the last period,P N i ¼1W iT ,as the objective function of the model.Instead,a sum of workload left at the end of each planning period or the total amount of delayed work,P T t ¼1P N i ¼1W it ,is used as the objective function.Constraints (1)maintain the crane flow,or movement conservation in each block when cranes are deployed from one period to the next.Constraints (2)ensure that only two cranes can serve a block in a planning period.These constraints have the following implication:•If there are already two cranes in a block at the beginning of a planning period,no crane can be moved into the block even after one or both of the cranes are moved out during the period.•If there is only one crane in a block at the beginning of a planning period,at most one crane can be moved in from other blocks even after the original crane is moved out during the period.•If there is no crane in a block at the beginning of a planning period,then at most two cranes from other blocks can be moved in during the period.These modeling implications match the practical operations.In a terminal,cranes should finish the work in blocks where they start and then be moved out to help others.Therefore,if one or both cranes can be moved out to help other blocks’work,it is not necessary for other cranes to move in to help that block.Constraints (3)maintain the balance between the workload that should be finished (W i ðt À1ÞþB it )and the workload that can be finished ðP N j ¼1Z ijt þP N j ¼1Y jit Þin each block by using slack variables W it .Constraints (4)ensure that the total workload fulfilled by a crane in each planning period cannot exceed its total net crane capacity.Constraints (5)initialize workload at the be-ginning of a planning horizon;any delayed workload from the previous planning horizon is counted as the workload in the first planning period.Constraints (6),together with the parameters X ii 0,define the initial locations of the cranes.Constraints (7)and (8)are non-negative and integer constraints.Constraints (8)also restrict that at most one RTGC can be moved from one block to another in a period.Recall that X iit is the number of cranes staying in block i during the whole period t and therefore it my take a value of 2.544 C.Zhang et al./Transportation Research Part B36(2002)537–555For each crane,if the total workload that it fulfills in block i and j plus its transportation time between the two blocks is less than its capacity in a period,the crane will be in an idle state for some time within the period either in block i ,block j or both.Based on the definition of Z ijt and Y ijt as shown in Fig.4,the crane can be moved from block i to j at any time in the range ðZ ijt ;C ÀY ijt Àt ij Þ.grangean heuristicAs indicated in Section 4.2,the model is a mixed-integer programming model.A common way to solve such a problem is to relax some coupling constraints and decompose the problem into several sub-problems that are easy to solve.See Section 5.1for the application of the method to our problem.Lagrangean relaxation has been successfully applied to problems in different settings,for ex-amples,Holmberg and Hellstrand (1998),Luh and Hoitomt (1993),Nilsson and Sjelvgren (1996),Rana and Vickson (1991)and Kim and Park (1998).Although Lagrangean relaxation is a heu-ristic method and cannot guarantee optimal solutions in general,it has the advantage of identi-fying the bounds of the optimal solution as part of the solution.The procedure suggests a solution and gives a measure of how well that solution approximates the optimal solution (Pirkul and Schilling,1998).Readers interested in the detail of Lagrangean relaxation are invited to read Fisher (1981)and Geoffrion (1974).grangean relaxation of the problemThe Lagrangean relaxation method is a heuristic method that first dualizes one or more con-straints to produce a Lagrangean problem,then solves the Lagrangean dual with subgradient optimization and at last finds a primal feasible solution by eliminating the infeasibilities in the dual solution.The objective value J of any primal feasible solution is an upper bound of the optimal objective value J Ãwhile the corresponding optimal dual objective value L Ãis a lower bound of J Ã.The difference between J Ãand L Ãis known as the duality gap.An upper bound of the duality gap is provided by J ÀL Ã,and ðJ ÀL ÃÞ=L Ãis thus often used a measure of sub-optimality of the feasible solution with respect to the optimal one (Luh and Hoitomt,1993).An approach like this can be quite efficient for a problem with a special structure.Below we describe a pro-cedure to solve problem CD_P based on the approach.5.1.1.Formulation of the dual problemThe capacity constraints (4)which combine the continuous and integer variables in model CD_P are relaxed by using the non-negative Lagrangean multipliers p ijt to form the relaxed problem:ðCD D0ÞMin LSubject to ð1Þ–ð3Þand ð5Þ–ð8Þ;C.Zhang et al./Transportation Research Part B36(2002)537–555545Edited by Foxit Reader Copyright(C) by Foxit Corporation,2005-2010For Evaluation Only.whereL¼X Tt¼1X Ni¼1W itþX Ni¼1X Nj¼1X Tt¼1p ijt½ðZ ijtþY ijtÞÀðCÀt ijÞX ijt¼X Tt¼1X Ni¼1W itþX Ni¼1X Nj¼1X Tt¼1p ijtðZ ijtþY ijtÞÀX Ni¼1X Nj¼1X Tt¼1p ijtðCÀt ijÞX ijt:This leads to two decomposed sub-problems:ðCD D1ÞMin L1¼X Tt¼1X Ni¼1W itþX Ni¼1X Nj¼1X Tt¼1p ijtðZ ijtþY ijtÞSubject to constraintsð3Þ;ð5Þandð7Þ:ðCD D2ÞMin L2¼ÀX Ni¼1X Nj¼1X Tt¼1p ijtðCÀt ijÞX ijtSubject to constraintsð1Þ;ð2Þ;ð6Þandð8Þ:Problem CD_D1involves only continuous variables and can be solved easily.Problem CD_D2 involves integer variables but it has a structure similar to the networkflow problem and can also be solved easily.Thus,for a given set of Lagrangean multipliers p ijt,problems CD_D1and CD_D2can be solved accordingly.The Lagrangean dual problem is:ðCD DÞMaxp P0L with L¼MinðL1þL2Þ¼Min L1þMin L2¼LÃ1þLÃ2Subject toð1Þ–ð3Þandð5Þ–ð8Þ:5.1.2.Solving the dual problemSince the sub-problem involves discrete variables,the objective function in CD_D is a piecewise linear concave function and may not be differentiable at certain points in the p space(Fisher, 1981).A subgradient method is therefore used to solve the dual problem(Held et al.,1974; Polyak,1969).In the algorithm,the multiplier p is updated iteratively according top nþ1 ijt ¼p nijtþh n D nijt;where n is the iteration number index,D nijt is the subgradient of L in CD_D with respect top;D nijt ¼ðZ nijtþY nijtÞÀðCÀt ijÞX nijt,h n is the step size determined by the rule given in Polyak(1969).h n¼kJÃÀL nPi¼1Pj¼1Pt¼1ðD nijtÞ;ð9Þwhere JÃis the best primal feasible solution found up to iteration n,L n is the optimal dual ob-jective value at the n th iteration,k is a parameter used to adjust the step size and usually k¼2 initially.When the dual solution does not increase for certain number of iterations,k is reduced by half.As proved by Polyak(1969),this method ensures the convergence of the Lagrangean dual solution at a geometric rate.546 C.Zhang et al./Transportation Research Part B36(2002)537–555。