Storage and stacking logistics problems in container terminals

Transcription

1 Original Article Storage and stacking logistics problems in container terminals Jiabin Luo a,yuewu a, *, Arni Halldorsson b and Xiang Song c a School of Management, University of Southampton, University Road, Highfield, Southampton, SO17 1BJ, UK. b Department of Technology, Management & Economics, Chalmers University of Technology, Vera Sandberg Alle 8, Gothenburg, Sweden. c Department of Mathematics, Logistics and Management Mathematics Group, University of Portsmouth, Lion Terrace, Portsmouth, P01 3HF, UK. *Corresponding author. Abstract In container terminal operations, the storage yard plays an important role for a terminal s overall performance because it links the seaside and landside and serves as the buffer area for storing containers. Therefore, storage and stacking logistics has become a field that increasingly attracts attentions in both academic and practical research during the recent years. The purpose of this article is to review and classify the growing literature on storage and stacking logistics, and to identify the research areas that could be further investigated. Theliteratureinthisareamainlyfallsintothreecategories:storage space allocation problem, design of optimal yard layout and container stacking logistics. This article is among the first known to review the literature focus on this area, it thus provides a new perspective for both managers and researchers on the issue of yard operations management. OR Insight (2011) 24, doi: /ori ; published online 13 July 2011 Keywords: container storage; container stacking; yard operations Received 10 November 2010; accepted 23 May

2 Storage and stacking logistics problems Introduction Brief description of container terminals Containers are standardised large steel boxes that were first used in the mid-1950s for transporting goods from one point to another. The proportion of cargo handled this way has been steadily increasing since then. Today, over 60 per cent of the world s deep-sea general cargo is transported in containers; however, some routes, especially between economically strong and stable countries, are containerised up to 100 per cent (Hulten, 1997). Container terminals serve as an interface between different types of transportation systems (see Figure 1). In today s highly competitive environment, most terminals in the world are working at or close to capacity; therefore, there is significant pressure to improve operational efficiency in container terminals. A container terminal operation typically consists of two parts: (1) the discharging (unloading) process of import containers, in which the containers are unloaded from a vessel, transported to the yard area and stored in the blocks, and (2) in the opposite direction, the loading process of export containers, in which the containers are loaded onto a vessel from the marshalling yard. The overview of container flows in the terminal is shown in Figure 2. Container storage systems The storage yard functions as a buffer area for storing various types of containers. The yard is divided into several blocks and each block is divided into rows, bays and tiers. Containers are stored in stacks in the yard (see Figure 3). Usually export and import containers are not stored in the same block, and special blocks are reserved for reefer containers and empty containers. Containers are reshuffled (re-handled) when containers located underneath have to be moved first because of incorrect information received or storage limitation of the yard. The storage logistics in the yard would decide Quayside Landside Stack with RMGC Vessel Quay Crane Vehicles Vehicles Trucks, Train Figure 1: Container transportation and handling chain in the typical port container terminal (Steeken et al, 2004). 257

3 Luo et al Truck Arrivals Landside Operations Pre-Gate System Transfer Areas Maritime Rail Terminal Container Storage Yard for Imports and Exports Shipside Operations Quayside Berths Ships Arrive/Depart Figure 2: Overview of container flow through a terminal. a stack a column a lane a tier a slot (position) Figure 3: Container storage blocks (Wan et al, 2009). individual storage slot for each container in order to minimise the reshuffles in the future. Yard cranes (YCs) are used for stacking containers in the storage yard. The most common types of YCs are chassis-based transporters, reach stackers, straddle carriers (SCs), rail-mounted gantry cranes (RMGCs) and rubber-tyred gantry cranes (RTGCs). Figure 4 displays the different cranes and their storage 258

4 Storage and stacking logistics problems CHASSIS REACH STACKER STRADDLE CARRIER RTGC Practical Storage Capacity TEU/hectare Figure 4: Different types of handling equipment in a yard ( accessed 2 January 2006). RMGC capacity. SCs, RMGCs and RTGCs are commonly used in container terminals. Among these equipments, SCs are not only able to transport containers, but can also lift and stack containers in the yard without the assistance of any other equipment. Only RMGCs are suitable for fully automated container terminals while RTGCs are more flexible in operation. Modelling techniques Most papers in the literature on solving storage and stacking logistics problems in container terminals are based on mathematical models, including linear programming, nonlinear programming (NLP) and mixed integer programming (MIP) models. Because of the computational complexity of solving these models to optimality, other techniques such as heuristic methods are used to solve these models to obtain an approximate (near-optimal) solution. A heuristic method is a procedure for tackling problems by an intuitive approach in which the structure of a problem can be interpreted and exploited intelligently to obtain a solution. Heuristics are usually applied to optimisation problems. Table 1 provides some background on the most commonly accepted techniques, which are used to solve problems in container storage and stacking area. Literature Review General overview of container terminal operations Container terminal operations are rapidly developing into a favoured research area by both academicians and practitioners. The increasing number of container shipments implies a higher demand on container terminals, storage 259

5 Luo et al Table 1: Commonly used heuristic methods for solving storage and stacking logistics problems Heuristic techniques Tabu search (TS) Simulated annealing (SA) Genetic algorithm (GA) Definitions TS uses a neighbourhood or local search procedure to iteratively move to find the best solution. A tabu list is used to store attributes of all the solutions. SA models as the physical annealing process for solids that are cooling down slowly to obtain a specified temperature level. The initial temperature is decreased as the process continues. GA operates on a population of potential solutions simultaneously, and combines the principle of survival of the fittest with structured but randomised information exchange to form robust exploration and exploitation of the solution space. References and examples Glover and Laguna (1998); Han et al (2008) Kirkpatrick et al (1983); Kang et al (2006) Goldberg (1989); Kozan and Preston (2006) management as well as on handling techniques (Steeken et al, 2004). New trends and topics in container terminal operations are discussed by Vacca et al (2007) where the authors point out that the combination and integration of problems could be investigated in the future. Generally, there are some specific operational issues that arise in the planning and scheduling of container terminals: such as berth allocation problem, crane assignment and split, stowage planning and sequencing, storage and stacking policies, and workforce scheduling (Günther and Kim, 2006). Among these, container storage has become a field of increasing importance because more and more containers have to be stored in ports because of the continuous growth of container traffic; the storage space is thus becoming a scarce resource (Steeken et al, 2004). Therefore, the efficiency of storage and stacking containers is one of the most important factors for a container terminal (Stahlbock and Vob, 2008). In the following sections, we summarise and analyse the recent literature on storage and stacking logistics. The optimisation problems in container storage and stacking area can be divided into three sub-areas: container storage space allocation problem, design of the yard layout and container stacking logistics. The optimisation problems related to these are listed in Table 2. Table 3 presents some of the studies in these three sub-areas from the literature. 260

6 Storage and stacking logistics problems Table 2: Typical optimisation problems in the area of container storage and stacking logistics Container stacking and storage logistics Storage space allocation problem Design of the yard layout Container stacking strategy Optimisation problems Determine the storage space of each block to locate different types of containers (export, import, transhipment, empty and reefer containers) of each vessel. Determine the layout type (parallel or perpendicular), the outline of the yard, the number of aisles between blocks/bays and the size of blocks. Determine the exact locations of containers in blocks for efficient use of storage space (location assignment problem); reduce the unproductive moves (container reshuffle problem); the choice of equipment. Container storage space allocation problem Storage location assignment is an important problem in warehouse management (Jane, 2000), and it is also an essential part of the container yard operations. In practice, in addition to blocks that are used to store the normal export and import containers, special blocks are usually reserved for reefer containers and empty containers. For example, empty containers are usually stored in a separate area. Reefer containers need to be stored in the blocks with the power equipment for maintaining the required temperatures; and this factor needs to be taken into account when allocating storage space of containers. Table 4 presents some key studies on the container storage space allocation problem. There is a recent trend to construct an integrated model to solve several of such problems simultaneously. The integrated problem of storage allocation and yard vehicle scheduling is first proposed by Bish et al (2001); a heuristic method is suggested by dividing this problem into two separate steps in which the first step consists of location assignments while the second step looks at vehicle scheduling. This work could be extended to a combined problem of determining every container s storage location, vehicles scheduling as well as quay cranes scheduling (Bish, 2003). On the basis of this work, Lee et al (2008) address an MIP for integrating yard truck operations and the container storage allocation problem. The objective is to minimise the make span of operations. Taking both loading and discharging requests into account, this problem has been further studied by developing a constructive heuristic-hybrid insertion algorithm by Lee et al (2009) where the objective is to minimise the total turnaround time of yard 261

7 Luo et al Table 3: Studies related to container storage and stacking problems in container terminals Container storage and stacking problems References Contributions Storage space allocation problem Jane (2000) Storage location assignment in warehouse management Chen et al (2003) Address the general yard allocation problem Zhang et al (2003) The first study to formulate the storage space allocation problem (SSAP) Bazzazi et al (2009) An extended version of SSAP based on the work of Zhang et al (2003) Lee et al (2006) Yard allocation problem in transhipment hubs Han et al (2008) Extend the work of Lee et al (2006)by assuming the yard design is unknown Nishimura et al (2009) Study the storage planning problem of transhipment containers in mega-containership Design of the yard layout Murty (2007) Brief discussion about the design and layout of a container yard Petering and Murty (2009) Study the effect of yard block length on the terminal operations Petering (2009) Study the effect of block width on the terminal operations Kim et al (2008) Both parallel and perpendicular layouts are considered Lee and Kim (2010) Estimate the block size with studying the locations of transfer points Stacking strategies Chen (1999) A study of unproductive moves Watanabe (1991) Propose a simple method for estimating the number of container re-handles Kim (1997) Develop a methodology to calculate the average number of re-handles Kang et al (2006) Identify stacking strategies for export containers with uncertain weight information Dekker et al (2006) Study a number of policies for stacking containers Hirashima et al (2006) Q-learning algorithm to determine the container movements in marshalling process Hirashima (2008) Extend the work of Hirashima et al (2006) to material handling problem Hirashima (2009) New reinforcement learning system for the marshalling plan to optimise the yard layout Kim and Bae (1998) Study how to move the export containers in order to reduce the time of loading process Lee and Hsu (2007) Consider the movements of individual containers within a bay during the marshalling process Lee and Chao (2009) Propose an alternative integer programming model to minimise the number of container movements during the marshalling process and to reduce the further re-handles simultaneously Integrated optimisation problems Kim and Kim (1998) Determining the optimal amount of storage space and optimal number of transfer cranes for the imports Kozan and Preston (2006) Simultaneously solving optimal storage strategy and container-handling schedule problems Bish et al (2001) Integrating the storage allocation and yard vehicle scheduling Bish (2003) Integrating storage location, vehicles scheduling and quay cranes scheduling Lee et al (2008) Integrating yard truck operations and the container storage allocation Lee et al (2009) Extending the work of Lee et al (2008)by considering both loading and unloading process 262

8 Storage and stacking logistics problems Table 4: Some key studies on the storage space allocation problem Authors Key findings/contributions Methodology Chen et al (2003) Zhang et al (2003) Bazzazi et al (2009) K Address the general yard allocation problem K Study a two-dimensional rectangle packing problem K The objective is to minimise the yard space used K The experimental results show that the GA approach can provide the best solution K The first study to formulate the storage space allocation problem (SSAP) K Investigate a complicated situation in Hong Kong s terminal, where all types of containers are mixed K The computational results show that the method proposed in their paper is very efficient for solving SSAP and also the computational time is reduced K An extended version of SSAP based on the work of Zhang et al (2003) K Reefer containers and empty containers are studied K Only consider the first level of Zhang s model Tabu search (TS), simulated annealing (SA), genetic algorithms (GA) and the squeaky wheel optimisation Rolling-horizon approach: the first level determines the total number of containers associated with each block in the yard; the second level determines the number associated with each vessel There are 22 numerical tests to be carried out to verify the performance of this extended model and GA trucks and the weighted sum of total delay. Computational results show that this method provides near optimal solutions to the problem. Lee et al (2009) also describe an efficient approach solving the loading and unloading operations in the container terminal. There are other models proposed in the literature integrating two or more problems. A model for estimating several cost components is developed by Kim and Kim (1998) to determine the optimal amount of storage space and optimal number of transfer cranes for the import containers. The model considers costs of space, trucks and cranes. A numerical example is provided in the article showing when space cost is increased the optimal amount of 263

9 Luo et al storage space will be reduced, which indicates there will be more containers stacked up in a bay. Conversely, increasing outside truck cost will increase the optimal number of transfer cranes as well as the optimal space amount. The optimal storage strategy and container-handling schedule problems can be solved simultaneously (Kozan and Preston, 2006). A novel iterative search technique based on genetic algorithms (GA), tabu search (TS) and a hybrid GA/TS algorithm is described for solving this integrated model, which includes a container transfer model and a container location model. Numerical tests on a real case in an Australian port are carried out. These show that the integrated iterative algorithm performs better than the individual models in stability. Overall, GA performs better than the hybrid technique. It is also found that reducing the maximum of container storage height results in the reduction of the ship s berth time. The major activity in transhipment hubs is the transhipment of containers, which means most containers unloaded from one vessel will be transferred to other vessels. Therefore, discharging and loading activities need to be considered simultaneously in planning yard operations. An MIP model is proposed to study the yard allocation problem by Lee et al (2006), where the objective is to determine the minimum number of YCs that can be deployed and the storage locations for containers. Furthermore, the reshuffling and traffic congestion can be minimised by the efficient transfer of containers between ship and yard. A sequential heuristic and a column generation heuristic were developed for solving the problem. The limitation of this work is that the authors assume that the yard design is given a priori, whereas in practice, yards may also need to be designed. However, in the literature, this is the first article that integrates the yard allocation problem with a consignment strategy and a vicinity matrix in transhipment hubs. Han et al (2008) extend the previous study to account for the locations of incoming containers as well. They formulate an MIP model to determine the storage locations, numbers of incoming containers and the smallest number of YCs to deploy in each shift. A high-low workload balancing protocol is used for reducing traffic congestion. The difference of this work to the work of Lee et al (2006) is that the yard design in this article is assumed to be unknown and has to be generated. Han et al (2008) describe a TS algorithm to obtain an initial yard template and develop an iterative improvement method based upon the sequential method for the yard allocation problem to solve the MIP model. Experiment results show that the proposed method can discover optimal solutions in most cases. One assumption of this model is that the reservation of each sub-block is fixed for only one vessel, which may not be the case in practice. The authors suggest that future work may address the problems related to changing reservation of blocks. 264

10 Storage and stacking logistics problems A recent work considering transhipment marine terminals is by Nishimura et al (2009). The contribution of this study is the consideration of the storage planning problem of transhipment containers in mega-containership (container ship with capacity of over TEU) at ports, which is different from previous research. They assume that the arrival pattern of transhipment containers is given in advance. An MIP model is proposed for the container storage allocation problem. A heuristic algorithm based on the Lagrangian relaxation is formulated and tested in accordance with mega-containership arrival rates, stacking strategies and terminal layouts. The results show that: the total service times for mega-containerships as estimated by this method are shorter than those estimated by other strategies; thus the solutions by the proposed method are very promising but need to be extended to take into consideration more realistic constraints, such as the weight class and destination port for each container. Furthermore, the stochastic model can be developed because the container stack situation and container storage volume in the yard area are changing over time. Design of optimal yard layout The block is the basic unit in the storage yard and can be broken down into three parts as bay, tier and row (see Figure 5). The width and length of the block are two important factors determining the yard storage layout. The size of a block can also affect the YCs productivity, thus influencing the overall performance of yard operations. There are two types of layouts considered in the literature: parallel layout in which blocks are laid out parallel to the berth; and perpendicular layout in which blocks are laid out perpendicular to the berth. tier row row bay Figure 5: The configuration of a container stacking block (Kim, 1997, p. 703). 265

11 Luo et al There are some studies in the field of yard layout design, which consider factors such as container handling equipments and a variety of costs. Murty (2007) gives a brief discussion about the design and layout of a container yard, but does not carry out any numerical tests. The author mentions that block size in the layout design of a terminal could affect the overall efficiency of the container terminal. The effect of yard block length on the long-run average quay crane rate is studied by means of simulation (Petering and Murty, 2009) where the authors find that gross quay crane rate is concave with respect to block length. Petering (2009) is the first in the literature to investigate the effect of block width on the terminal operations. This work investigates the yard with a parallel layout and draws a similar conclusion to the above work of Murty (2007): By assuming yard storage capacity and the yard equipment numbers are constant, the quay crane rate is concave with respect to the block width. The numerical experiments are carried out based on a fully integrated, discrete event simulation model. In the work by Kim et al (2008), another method for designing the layout of container yards is presented. In this work, transfer cranes are used for stacking containers in the yard, and yard trucks are employed for delivering containers between the quayside and yard side. Both types of layouts are considered. Several formulas are provided to evaluate the expected travel distance of trucks and expected number of relocations for transfer cranes. The numerical results show that in a parallel layout yard, the expected travel distance is shorter as compared to a perpendicular layout. Furthermore, the sum of travel cost and relocation cost is lower in the parallel layout. However, this study assumes that the yard is rectangular and the gate is located in the middle of the yard, which is an ideal assumption and may be different in practice. Lee and Kim (2010) have looked at estimating the block size at a container terminal. Two types of transfer points locations were analysed: transfer points besides each bay within blocks and transfer points at both ends of each bay within blocks. The results from a numerical experiment show that the optimal number of bays in blocks with a transfer point at each side of a bay was larger than that in blocks with transfer points at the ends, while the optimal number of rows in blocks with a transfer point at each side of a bay was smaller than that in blocks with transfer points at the ends. The results also show that when increasing speeds of the trolley and the gantry, the optimal number of rows and bays will be increased separately. Container stacking logistics The objectives of a stacking strategy are to use the storage space efficiently and minimise the number of container reshuffles. A reshuffle (re-handle) is 266

12 Storage and stacking logistics problems defined as an unproductive move of a container, which locates on top of the required one. A comprehensive discussion about the management of yard operations in the container terminal is provided by Chen (1999). By discussing the exports and imports storage management planning separately as well as analysing the storage strategies, a conclusion can be drawn that the higher container stacking in the yard would result in an increase in the number of unproductive moves; the impact of these moves could be reduced by improving all other relevant conditions simultaneously, such as good quality of container information received, improved storage strategies and so on, and therefore reduce the operational cost and increase the terminal efficiency. Early work on estimating the number of container re-handles was investigated by Watanabe (1991), who suggests a simple method, called an accessibility index, as an estimation of the number of re-handles. Subsequently, Kim (1997) further develops a methodology to calculate the average number of re-handles for the next container to be picked up and the total number of re-handles for picking up all the containers given an initial layout. The author studies several stack configurations and their effects on the number of re-handles and finds that the total number of container re-handles is related to the stack height more than its width. This method outperforms the Watanabe s method in both accuracy and unbiasedness. A simulated annealing (SA) heuristic has been developed to identify stacking strategies for export containers with uncertain weight information in Kang et al (2006). In this work, the authors estimate the probability distribution of each weight group and the number of container re-handles. Numerical tests show that this strategy performs better in reducing the number of re-handles as compared to the traditional same-weight-group-stacking strategy. It is also shown that using a machine learning algorithm could increase the accuracy of the weight classification. In the scientific literature, stacking problems can be dealt with in two ways: simplified analytical calculations or detailed simulation studies (for example, Dekker et al, 2006). In the work of Dekker et al (2006), they study a number of policies for stacking containers in a yard through simulation method at an automated container terminal. An overview of stacking policies in both academic work and in real-life operations is provided. In their work, they discuss random stacking and category stacking separately, where the classification of categories is similar as those used in the stowage planning: the weight, container destination and the type of a container. In this article, they consider several variants of category stacking and assume that containers can be exchanged during loading process. It is concluded that the capacity of a set of automatic stacking cranes is exceeded by the super containerships 267

13 Luo et al and the category stacking shows a better performance than random stacking. They also study the workloads of the cranes used for stacking and find that the peaks in the workload of automated stacking crane can be reduced by adding a workload control variable and by stacking on piles close to the transfer points. Marshalling is an approach that locates the containers in the order associated with the loading sequence before loading onto the ship; it is therefore one way to reduce the number of re-handles while loading. This process consists of two stages: selecting a container to be rearranged and removing the containers on the selected container in the previous stage. Only a few studies can be found in the literature on optimising the marshalling process in the yard. A new Q-learning algorithm is proposed by Hirashima et al (2006) for determining desirable movements of containers so that the total turnaround time of ships can be reduced. The desired layout in a buffer area is arranged according to the loading sequence of a vessel and each container has several desired slots for improving the learning performance. This Q-learning algorithm is based on the number of container movements for material handling in the yard. One finding from the numerical experiments is that the number of movements generated in this method is much smaller than the number generated by non-mathematical methods. The authors also suggest that the number of container movements can be reduced to obtain a desirable layout in the case of large-scale problem. Hirashima (2008) extends the Q-learning algorithm to the material handling problem. This study differs from the previous work in that the desired positions of each container are based on the shipping order in order to improve the learning performance. The results show that this method performs well in determining desirable movements of containers for reducing the ships turnaround time. A new reinforcement learning system for the marshalling plan has been proposed by Hirashima (2009) to optimise the yard layout, determining the order to move containers while considering container destinations at the same time. There are two stages in the proposed model: the first one is to determine the rearrangement order, the second one is to select the destination for removal containers, and the stages repeat sequentially according to container movements. A new learning model considering container-groups and corresponding Q-learning algorithm is used to determine the schedule of container movements. Markov s decision process is used to describe the layout and container movements. Simulation results show that the proposed model could generate solutions that result in a smaller number of container movements. The problem of how to move the export containers in order to reduce the time of loading process can be described into three sub-problems (Kim and 268

14 Storage and stacking logistics problems Bae, 1998): bay matching problem, move planning problem and task sequencing problem. The bay matching is to match a specific bay with a bay in the target layout; move planning problem is to determine the number of containers to be moved from this specific bay to another; then the completion time of the marshalling process is minimised by sequencing the tasks. Bay matching and task sequencing problems can be solved by dynamic programming techniques while move planning problem is addressed using the transportation problem technique. Kim and Bae (1998) suggest developing heuristic techniques for computational efficiency. However, they only consider container movements between bays, while movements within the same bay are not included. On the basis of this work, Lee and Hsu (2007) take into consideration the movements of individual containers within a single bay for the container marshalling problem, and develop a mathematical model, which is based on a multi-commodity network flow model. They focus on yards that use RMGC as its major container handling equipment. The aim of the problem is to minimise the number of movements required to transform the container yard from its initial layout to its final layout. The authors mention that the proposed model can be extended by taking into account the following three different requirements on the final layout: (a) the location of each container is specified at the final layout; (b) each stack holds just one container type; and (c) allow container to leave the yard at the same time as marshalling takes place. The model can also be simplified for computation efficiency by allowing multiple containers be moved in a one-time segment. However, only a single ship is considered in this work. Lee and Chao (2009) propose an alternative integer programming model to minimise the number of container movements during the marshalling process and to reduce the further re-handles simultaneously. The authors develop a heuristic method to generate an efficient sequence of container movements. This heuristic method takes a neighbourhood search process, which starts from a feasible solution and improves the solution by iterations. The solution process includes five subroutines: The first major subroutine decides the movement sequences for a good final bay layout with possibly a longer sequence. In the second major subroutine, a binary integer program is used to minimise the length of the sequences and helps preventing conflicts. The other three subroutines are minor subroutines, which are stack emptying subroutine, sequence reducing subroutine and mis-overlay index reducing subroutine. The authors focus on yards that use RMGC and only consider marshalling containers in one bay. Computational tests are carried out to demonstrate the performance of the heuristics as well as the effects of the subroutines. 269

15 Luo et al Conclusion Storage space is an important resource, the utilisation of which determines the overall performance of the container terminal. In this study, we review the literature about the storage and stacking logistics, which include container storage space allocation problem, design of the block size, design of the yard layout and container reshuffle problems and so on. The process of determining storage locations usually consist of three stages: allocating storage space for a set of containers, selecting individual storage slots and storing the reshuffled containers. Furthermore, the length and width of blocks are important variables in designing the layout of yard and they also influence the container handling process. Generally, decision-making problems for locating and reshuffling containers are the main challenges in this area. Because container reshuffle takes a significant amount of time, the most difficult problem here is the estimation of the number of reshuffles when retrieving containers and reducing the number of reshuffles. For example, as the information of container weight is unknown until the time when containers actually arrive, the number of reshuffles can be reduced by improving the accuracy of estimating the weight distribution because light-weighted containers are usually stacked on the heavy ones. It is also interesting to find that there is some recent work using game theory and pricing policy to solve container terminal problems. For example, Saeed and Larsen (2010) propose a two-stage game, which is based on two basic concepts of the cooperative game: characteristic function and core, to find the possible combinations within container terminals at one port; optimal pricing policies for solving storage and space allocation problems can also be found in, for example, Holguin-Veras and Jara-Diaz (1999, 2006) and Kim and Kim (2007). Further research on the issue of storage and stacking logistics could proceed in several directions: (1) Developing methods to find near-optimal solutions. For example, effort could be devoted on using meta-heuristics approaches, like TS, SA, GA, to solve the space allocation problem: more experiments could be conducted on large-sized instances using heuristic methods for finding feasible solutions as the models can only be used to optimally solve for smallsized cases using off-the-shelf optimisation software. (2) Special cases could be extended to take into consideration more general and realistic situations: for example, loading and unloading operations can be considered simultaneously with multiple quay cranes involved. MIP models are commonly used to formulate such problems. (3) Additional factors, such as traffic conditions, cost values, yard truck/cranes performances and the maximum yard depth can be taken into account for finding an optimal layout of a yard. As an example, the increasing volume of containers and multi-level stacking can cause traffic 270

16 Storage and stacking logistics problems congestion, and studies could be carried out on reducing the traffic congestion in a yard to minimise unproductive reshuffles of containers. Another interesting field of research is to study the relationship between the yard layout and the overall performance of container terminal. Simulation or mathematical models could be employed for this purpose. Because operations in container terminals run synchronously, optimising a particular aspect of the system cannot guarantee the improvement in the overall productivity of container terminal operations. Integrated scheduling of container handling equipment and container storage is therefore essential in improving the efficiency of container terminals. Information technology has been introduced in container terminals in recent years and it becomes an essential part in transferring large volume of data rapidly. Kia et al (2000), for instance, develop a simulation model to compare the productivities of two terminals with different operational systems; whereas one terminal uses information technology, no such system is used in the other. Current trends in maritime freight transport and shipping are analysed by Mangan et al (2008). Panayides and Song (2008) make a first attempt to provide theoretical and empirical measures of seaport terminal supply chain integration. In a subsequent work, the same authors, Panayides and Song (2009), define and empirically develop measures for container terminal supply chain integration. Dynamics and uncertainties always exist in container terminal systems. For example, the actual arrival time of a ship often varies from the scheduled time because of unexpected weather conditions and the transportation conditions from the terminal the ship departs from. Sometimes there is incorrect or incomplete information on containers, which would affect future container locations in the yard area. Stochastic programming models can be applied to handle uncertainties in the decision-making process. Higle and Sen (1999) initially present the idea that stochastic programming optimises an expectedvalue criterion, and it often includes constraints on downside risk. However, for the topics related to containers, the stochastic programming approach has been only applied for solving problems of empty containers allocations (see, for example, Crainic et al, 1993; Cheung and Chen, 1998; Leung et al, 2002). To date, no work has been done on using stochastic models for container storage problems in container terminals. About the Authors Jiabin Luo is a PhD student in the School of Management, University of Southampton working on optimisation problems at container terminals. She 271

17 Luo et al obtained a BSc in Mathematics from Zhongshan University in China in 2005, followed by an MSc in Statistics from University of Nottingham in 2006 and another MSc in Risk and Stochastic from London School of Economics in Yue Wu is currently a reader in the School of Management, University of Southampton (UK). She received two PhDs in the field of Operational Research at the Department of Management, London School of Economics (UK) in 2010, and in the field of Control Theory and Control Engineering at the Department of System Engineering, Northeast University (China) in Dr Wu had been a senior research associate at the Department of Management Sciences, City University of Hong Kong. Her research interests are global supply chain and logistics, port operations, air cargo forwarding, empty sea container allocation, stochastic modelling and NLP. Arni Halldorsson is a professor in supply chain management at Chalmers University of Technology, Sweden. Before that he held positions as senior lecturer and academic director of MBA at University of Southampton UK, and associate professor at Copenhagen Business School, Denmark. Key research topics include logistics, supply chain management, buyer supplier relationships and environmental sustainability, but his list of publications also reflects a strong interest research methodology. Professor Halldorsson has published his work in a number of leading academic journals within his discipline, has acted as co-editor of two books and of special issues of academic journals. Xiang Song is currently a lecturer at the Department of Mathematics, University of Portsmouth. She has been working on cutting and packing problems since 2001 and has extensively used mathematical programming and meta-heuristics techniques. References Bazzazi, M., Safaei, N. and Javadian, N. (2009) A genetic algorithm to solve the storage space allocation problem in a container terminal. Computers & Industrial Engineering 56: Bish, E.K. (2003) A multiple-crane-constrained scheduling problem in a container terminal. European Journal of Operational Research 144: Bish, E.K., Leong, T.Y., Li, C.L., Ng, J.W.C. and Simchi-Levi, D. (2001) Analysis of a new vehicle scheduling and location problem. Naval Research Logistics 48: Chen, P., Fu, Z., Lim, A. and Rodrigues, B. (2003) The general yard allocation problem. Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-03), Berlin, Heidelberg, New York, Springer-Verlag, pp

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