Efficient Container Security Operations at Transshipment Seaports

Transcription

1 1 Efficient Container Security Operations at Transshipment Seaports Brian M. Lewis, Alan L. Erera, and Chelsea C. White III Abstract This paper describes an approach for aiding the management of a container transshipment seaport to decide on the balance between the percentage of containers to undergo security inspection and the concomitant departure delays of outbound vessels. Security concerns heightened by the events in the United States on 11 September 21 have resulted in the U.S. Customs Service seeking partnerships with non-u.s. ports so that some portion of containers bound for U.S. ports might undergo security checks prior to U.S. entry. However, delays in outbound vessel departures can reduce a port s competitive position. We use a best-first heuristic search procedure, A*, to model the problem of (1) moving containers from inbound ships to staging areas, areas where security inspections can occur, and outbound ships and (2) moving containers from staging areas and areas where security inspections have been completed to outbound ships. It is assumed that the load plans for the incoming and outgoing vessels are known and that the containers to be inspected are known prior to inbound ship arrival. We discuss relevant modeling details for the initial model formulation, and discuss extensions of the problem. Index Terms Freight transportation security, intermodal freight, seaport container management. I. INTRODUCTION he health of the world economy today depends on an Tefficient, reliable global freight transportation system. Recent events have heightened concerns that the international freight system is vulnerable to exploitation or disruption by criminal and terrorist groups. The value of export goods produced and transported globally is staggering: $6.186 trillion in 2 [1]; clearly, the economic impact of a system disruption could be quite significant. To partially address Manuscript received August 15, 22. This ongoing research reported in this paper is supported by The Logistics Institute Asia Pacific, a research partnership between the Georgia Institute of Technology and the Republic of Singapore, and the ATLANTIC Project, a research partnership between the ITS Joint Program Office of USDOT, the Canadian Transportation Ministry, and the European Commission. B. M. Lewis is with the School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA USA ( blewis@isye.gatech.edu). A.L. Erera is with the School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA USA ( alan.erera@isye.gatech.edu). C C. White III is with the School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA USA (phone: ; fax: ; author@nrim.go.jp). these concerns, the United States Customs Service announced the Container Security Initiative (CSI) in early 22 [2]. Maritime transportation is the dominant mode for international trade, and 8% of the goods (measured by value) moved by ocean are transported in intermodal containers [3]. One of the key components of CSI is for participating non-u.s. ports to provide security inspection capabilities so that certain containers bound for the United States could be screened and pre-cleared for importation. Specifically targeted for participation in this initiative are the large transshipment megaports that serve as regional hubs for containerized cargo; the Port of Singapore has already agreed to participate. By distributing container security procedures globally, CSI intends to allow more high-risk containers to be identified and checked than is currently viable at U.S. ports of entry. Although the U.S. is currently the only importer that has developed an initiative for distributed container inspections, other nations may follow. In general, world demand for increased security in the intermodal container system is growing. To participate successfully in CSI or to provide security inspection as a value-added service to increase competitive advantage, a transshipment port needs to integrate new container security procedures into existing port operations without significantly impacting productivity. The time between the arrival of inbound vessels and the departure of outbound vessels is a key productivity measure for a transshipment port. Keeping this time to a minimum is an important port objective. However, container inspections take time and hence can affect turn-around time. A key question that faces port management is: What percentage of containers can be inspected while keeping outbound ship departure delay as small as possible? Clearly, departure delay is dependent on a variety of factors, including: The efficiency of the port in moving containers from inbound vessels to outbound vessels, usually via staging areas in the port. The positions of containers on inbound and outbound vessels (the stow plans) The number of containers to be inspected, and the locations of these containers in the stow plans. The objective of this research is to aid port management in deciding how to balance the percentage of containers to check without unduly increasing outbound ship departure times. To

2 2 do so, we begin by developing a set of problems in optimal transshipment container management that explicitly account for security procedures while minimizing ship turn-around time. We then focus on a simple variant with sufficient realism that can be modeled as a shortest path problem. II. INSPECTION AT TRANSSHIPMENT PORTS Regional transshipment seaports are the hubs of the international intermodal container system. Vessels arrive at such a port loaded with containers that need to be transshipped by other vessels to different local, regional, or global seaports. Through a sequence of container moves by cranes and tractors, the transshipment containers will be unloaded from inbound vessels, stored temporarily in stacks on the port apron, and eventually loaded onto outbound vessels according to preexisting stow plans. We define a container move to be a transfer of the container from one point-of-rest to another point-of-rest. Suppose now that some of the transshipment containers require a security inspection procedure prior to being loaded on an outbound vessel as part of a mandated security policy or as a security service. For most procedures, containers to be inspected will need to be moved into a dedicated area of the terminal yard for the procedure. As an example of the sequence of events in this process, customs agents representing the nation of importation will analyze a vessel manifest data and target a subset of containers to be inspected, preferably prior to arrival. Upon arrival, the container unloading process begins. When a targeted container is unloaded, it may be moved directly to the inspection area, or to a yard stack if no space is available. The inspection process itself may be nonintrusive, intrusive, or both. Non-intrusive inspections most commonly utilize x-ray and/or gamma-ray based screening systems as well as radiation detector pagers. If the results of the non-intrusive inspection are not satisfactory to clear the container for import, the container may be opened and stripped by agents. Once inspection procedures are completed for a container, it may be moved directly to a slot on an outgoing vessel, or to a yard stack if its departure slot is not yet available. To determine the impact of inspecting an increasing percentage of transshipment containers on vessel departure times, this research begins to develop a set of transshipment port container operations models that explicitly include security inspection processes. Our initial approach is to formulate models that, when solved optimally, would provide the sequence of container moves (including the order in which inspections take place) that lead to minimum departure delays, and thus maximum port efficiency. III. SUITE OF CONTAINER MANAGEMENT PROBLEMS Most port-related operations research has focused on the development and analysis of models for vessel stow plan design, terminal yard storage planning, berth allocation, and equipment routing and scheduling. Furthermore, these problems have been treated individually rather than as components of an integrated port container management model. Apparently, there is no port operations literature that considers security and its relationship to efficiency. In this section, we begin by describing a highly simplified problem that addresses integrated container management with a security component. The formulation, which we denote the base case, is intended to be a conceptually and computationally simple starting point from which more realistic problems can be developed. We then discuss various ways in which the base case can be realistically extended. Many of these extensions are topics for future research. A. The Base Case Assume that M fully loaded inbound vessels and N empty outbound vessels berth simultaneously at a transshipment seaport. The inbound vessels are laden with a set of containers, all to be transshipped via the outbound vessels. A subset of the containers must complete an inspection at the security station prior to being loaded onto their outbound vessels. The inspection time is assumed to be known and deterministic, and each inspected container is permitted to continue in the transshipment process once its inspection is complete. The list of containers that require security inspections, which we denote the inspection list, is assumed to be known at the beginning of the transshipment process. Finally, we assume that the location of each container is known at all times. The problem objective is then to determine the serial sequence of container moves that completes the transshipment process for all containers, including any required inspections, in the minimum time. Define a container move to be a transfer of a container from one point-of-rest directly to another point-of-rest. A point-ofrest may be a specific stowage slot on a vessel, or a position in the terminal yard where a container can be stored or inspected. Yard storage of containers in large ports is organized into container stacks. These stacks are organized piles of containers, possibly with established widths (x), lengths (y), and heights (z) given in number of containers. Any container slot in a specific stack can be identified by the triple coordinate (x,y,z); see Figure 1. Yard stacks may be dedicated to a specific outbound vessel (a ship stack), or may be comprised of containers destined for multiple outbound vessels (a universal stack). In the base case, we assume that the terminal yard has a single universal container stack for temporary container storage during transshipment. Further, we assume that the yard contains a single distinct area dedicated to container inspections (the security station). In the base case model, we assume a limited number of feasible container moves. A container can be moved from its starting location on an inbound vessel directly to its stowage slot on its outbound vessel, to a slot in the universal container stack, or to the security station. From a slot in the ship stack, a container can be moved to its stowage slot on its outbound vessel, to the security station, or to another slot in the ship

3 3 stack (a shuffling operation). Finally, a container in the security station can be moved to its outbound stowage slot or to a slot in the container stack, but only after its security height width length FIGURE 1 Three-dimensional view of a container stack inspection has been completed. In addition to these restrictions, container moves must also obey certain obvious physical restrictions. For example, a container cannot be moved to a destination slot unless there is already another container stored in the slot immediately beneath that position. We assume that containers in the security station are processed by a single inspection server in first-in first-out (FIFO) order. The solution method developed however can be extended to cases with multiple servers and other queuing disciplines. B. Extensions to the Base Case 1) Inspection list determination timing A key base case assumption is that the inspection list is known by the beginning of the transshipment process. This may not always be the case in practice, and certain containers may be flagged for inspection after the transshipment process has started. One topic for future research is to understand the impact on efficiency of the difference between the time of the start of the transshipment process and the time when the inspection list becomes completely known. 2) Modeling parallel operations within seaports Another assumption in the base case is that container moves are performed in serial, e.g. the next container move cannot be initiated until the previous container move has completed. The loading and unloading of vessels at a seaport is actually a complex parallel process with multiple quay cranes, yard tractors, rubber-tired gantry (RTG) cranes, and other equipment working simultaneously. As a result, we intend to explore in the future parallel transshipment modeling as an extension to the simpler serial case. 3) Dynamic ship arrival process The base case problem also treats the ship berthing process during transshipment in a highly simplified manner. Rarely are a set of empty out-bound ships awaiting the arrival of loaded in-bound ships before transshipment begins; more realistically, in-bound ships become out-bound ships are they are unloaded and transshipment is a ongoing process. Ship arrival time is a function of scheduled arrival time and a variety of other factors that often can be reasonably modeled using a random variable for arrival delay. Modeling this dynamic process of ship arrivals is an important extension to our base case problem. 4) Mixed transshipment and import-export processes We intend also to consider an extension to the base case in which not all containers undergo transshipment. Some outbound containers may already be stored in the container stacks at the start of the transshipment process, and some may enter the yard during the transshipment process. In addition, a subset of the unloaded in-bound containers may remain in the container stacks after the outbound vessels are completely loaded. Again, this extension may increase realism and provide better estimates as to how security processes impact efficiency. 5) Secondary inspections In the base case, the security procedure is modeled as a single-phase inspection which all containers pass. This assumption is reasonable; if a container did not pass and required a time-intensive secondary inspection, it is likely that this procedure would be conducted in parallel with the remaining primary inspections. We also assume in the base case that every inspected container eventually will be stowed on one of the out-bound ships. In reality, the outcome of the inspection process may remove a container from, and hence alter, an out-bound stow plan. Formulations that explicitly consider the impacts of secondary inspections and the possibility of altered stow plans would be useful extensions to the base case formulation. 6) Value of information technology The base case implicitly assumes that the seaport has an information technology (IT) system that tracks the location of each container and piece of equipment at all times. We can explore the value of knowing this real-time information with the following problem variant. We still assume knowledge of the location of each container on the vessels and in the security station, but not within the container stacks. Each time a container is to be retrieved from the container stacks, rather than incurring the time to travel directly to the appropriate storage area and slot, we assume instead that the system incurs a stochastic search time to find and then travel this location. The Port of Singapore is one of the world s largest seaports in terms of both twenty-foot equivalent units (TEUs) and gross tonnage handled [4]. The Port of Singapore processes the most transshipment containers annually among world ports, and is widely considered one of the most efficient. High-level operations personnel at the port claim that IT is almost a necessity for handling port container throughput [5]. Kia, Shayan, and Ghotb [6] discuss the results of a simulation model that shows the benefits of using certain information technologies in seaport terminal operations. One of the main reasons for the improvement in operations reported was total visibility of the system and thus the elimination of the search

4 4 time required to find a container in the container stacks. C. Modeling Security and Efficiency Understanding the trade-off between security and efficiency is important both to port operators and users. If vessels incur departure delays due to inefficient operations because certain containers are waiting to obtain customs clearance or to be inspected, the shipping lines may move their business to other ports [5]. However, ports that are not considered secure may be at a significant competitive disadvantage. Consider the following approach for quantifying the impact of security on port efficiency. Let τ(s I, S O, S S ) be the minimum time required to unload in-bound ships with stow plans S I and load out-bound ships with stow plans S O, assuming that each container in the set S S is to undergo a security inspection. We remark that determination of τ(s I, S O, S S ) is an as yet unsolved optimization problem on which our research efforts are focused. Given S I, S O, and S S, security inspection delay is Conditional Cumulative Probability P ρ (α) FIGURE 2 Delay α P ρ (α) for a Fixed Value of ρ (S I, S O, S S ) = τ(s I, S O, S S ) - τ(s I, S O, φ), (1) where φ is the null set. We assume that the cumulative distribution function over S I, S O, and S S is known and given. Let A be the number of elements in (i.e., the cardinality of) the set A. Throughout, we assume S I = S O ; hence, all containers on the in-bound ships are placed on the out-bound ships and all containers placed on the out-bound ships were placed on the in-bound ships. We seek the following conditional probability for all α: P ρ (α) = P[ (S I, S O, S S ) < α ρ = 1* S S / S I ], (2) which is the probability that the out-bound ships will be delayed no more than α units of time, given the percentage of containers to be inspected is ρ. Figure 2 presents a graphical depiction of this function, for fixed ρ. We conjecture that for all α, P ρ (α) is non-increasing in ρ. An implication of this conjecture, assuming this conjecture is true, is that the higher the percentage of containers to be inspected, the higher the expected delay will be. This implication is graphically depicted in Figure 3. Figure 3 can aid a port manager in deciding what percentage of containers to undergo security inspections and how to price the new service. IV. BASE CASE FORMULATION Since most of the optimal container management problems proposed earlier in this paper are very difficult to analyze and solve optimally, we first address developing a solution procedure for the serial operations base case. Although this problem is simplistic since it assumes serial container move sequences and simultaneous vessel arrivals, it differs from previous research in this area in that the entire transshipment process within a port is treated in an integrated model. Most existing research of container port operations treat unloading Expected Departure Delay (in minutes) Inspection Level (%) FIGURE 3 Percentage of Containers Inspected vs. Expected Departure Delay and loading processes [7,8] and equipment scheduling [9,1] as individual problems rather than as a single systematic problem which occurs in practice. The following section of the paper presents the shortest path problem formulation for the base case and the determination of τ(s I, S O, S S ). We assume M inbound vessels and N outbound vessels are simultaneously berthed at the seaport to transship a set of containers. The terminal yard consists of a single universal container stack and a security station at which one-phase container inspections of deterministic duration occur. Let B be the set of all storage areas, e.g. all vessels, the container stack, and the security area. Each storage area is assumed to be threedimensional and rectangular in shape with defined width, length, and height. The security station can only handle a single container at any time. The M in-bound vessels need to unload a set of containers, and the position of each container in the in-bound stow plans is

5 5 known. Containers will be loaded onto the outbound vessels according to the known out-bound stow plans which detail the required assignment of containers to outbound vessel storage slots. Since all vessels arrive simultaneously and no containers arrive except on the in-bound vessels or depart except via the out-bound vessels, we know that each storage slot in the terminal yard is unoccupied at the start and at the end of the transshipment process. A subset of the containers being transshipped will require a security procedure prior to being loaded on an outbound vessel as part of a mandated security policy or as a security service. Let X i be an binary variable that indicates whether container i { 1,2,... U} is targeted for inspection (X i =1) or not (X i =). When a container has been targeted for inspection, the container is required to be moved into the security station at some point prior to being loaded, and can be removed only after the completing the inspection procedure. Containers that are not targeted for inspection cannot enter the security station at any time. Let I represent the inspection status of all containers being transshipped. For i { 1,2,... U}, let I[ i] = 1 if container i has completed a security inspection and zero otherwise. At the start of the transshipment process we have not inspected any containers, and therefore I[ i] = for i { 1,2,... U}. A. Notion of System State We consider the state of the system to be the contents of all storage slots in the system, the location of each piece of equipment, and the inspection status of each container. Specifically, let n = { W, I} be a state and let N be the set of all feasible states. In the base case, we know a priori the starting state, n o = { Wo, I o}, and the state that marks the end of transshipment process, the goal state, n = W, I }. γ { γ γ B. Defining the Network Let each state n N be represented by a node in a network and the set of nodes is N. For state n N, we define the successor set of n to be the set of states which can be reached from state n using a single container move. We refer to the creation process of this set as successor set generation or node expansion, and denote the set by SCS(n) [11]. We also denote the branching degree of n as b ( n) = SCS( n). Let a directed arc ( n, n ) exist between nodes n, n N such that n SCS(n). Let A be the set of all such directed arcs, and define the network G = ( N, A). Since the terminal system includes a finite number of containers to be transshipped, U, and a finite number of storage locations, clearly N is a finite set and so for n N, we have b (n) <. Therefore G is a locally finite graph. A path, p i,j, through the network is defined as the sequence of nodes n i, n i+1, n j such that n t SCS( nt 1 ) and the set of all paths from n i to n j as P i,j. A node n j is said to be accessible from node n i if a path is exists from n i to n j. A path then represents a serial sequence of feasible container moves. We assign a cost to each arc a = ( n, n ) A and denote it c(a) or c ( n, n ). The cost is the time to complete the single container move. This move time will be composed of three components: set-up time, travel time, and placement time. The set-up time is composed of four components: the time for the crane, an RTG or quay crane, to move from its current position to the (x,y)-coordinate of the container to be retrieved; the time for the crane to lock onto the container; the time for the crane to position itself over the (x,y)-coordinate of the new storage slot or over yard tractor; and finally the time for crane to unlock the container. The container is at rest in the stack or on the yard tractor (ready to be moved). The travel time is simply the time for the yard tractor to drive from its loading point to its unloading point. Both points may be at the container stack or under a quay crane. The placement time is essentially the reverse of the set-up time and has four components: the time for the crane to move from its current position to over the yard tractor; the time for the crane to lock onto the container; the time for the crane to position itself over the (x,y)-coordinate of the new storage slot; and the time for the crane to unlock the container. Clearly, for shuffling moves within the universal container stack, there is no travel time or placement time. We define the cost of a path to be the sum of all arc costs along the path. The cost of path p i,j is given by j 1 i, j ) = c( nt, nt+ 1) t= 1 c ( p (3) If a container has been moved into the security station for inspection, we do not simply add the deterministic inspection time to the move time to create the move cost. This would prevent all other container moves from occurring until the inspection was complete. During the time required to complete the inspection, container moves involving the container undergoing inspection are ruled out as infeasible. This is accomplished using path cost logic. Denote the inspection time as InspTime and assume that the container was moved into the inspection area to form node n insp. Let the path from the start node no to cost be p o, insp ninsp with the minimum known. We then constrain the successor set generation process for all nodes accessible through n insp as follows. Consider a node n k that is accessible through n insp. If inequality k 1 t= insp c ( n, n InspTime (4) t t+ 1 ) holds at node k n, then the movement of the container in the inspection area is considered a feasible container move in

6 6 generating the successor set for node n k ; otherwise it is not. This technique also allows the modeling of security areas with more than one inspection channel, and a variety of deterministic queuing disciplines. C. Shortest Path Problem We define the shortest path to be the path p equations 5 and 6 are satisfied. γ 1 ( p ) = t= + 1 ) * Po, γ such that c c ( n, t n t for path p P o, γ (5) p* = arg min{ c( p)} (6) p P o, γ The shortest path describes the optimal serial sequence of container moves to complete the transshipment process. V. SOLUTION TECHNIQUE One of the major issues in modeling a complex system like a container transshipment port is that the system may include thousands of transshipment containers and thousands of container slots. This level of detail leads to an extremely large state space and thus node space. For that reason we implement the A* optimal algorithm for the shortest path formulation. This method is especially useful in this case since it may prune entire portions of the network based on a node-selection function and thereby eliminate the exhaustive network searches used in many dynamic programming techniques. (See pages in [11] for a description of the A* algorithm.) Pruning in this way results in large portions of the state space never explicitly generated in the solution procedure. A* is complete and admissible on finite networks, and thus terminates with a solution if a solution exists and is guaranteed to terminate with an optimal solution if an optimal solution exists [11]. The A* algorithm is based on the idea of generating successor sets for nodes that seem to be on or close to the shortest path to the goal node. For each node we define a node-selection function, f*, which is composed of two components, g* and h* such that f* = g*+h*. For n N, g*(n) is the minimum cost of all paths from the start node to node n whereas h*(n) is the minimum cost of all paths from node n to the goal node. Of all nodes that can be expanded, A* chooses the node with minimum node-selection function value. The problem is that this information is generally not known during implementation. If it was known, A* would only expand nodes along the shortest path from start node directly to the goal node. f* is known as a perfect discriminator since it would detect any node that was not on the shortest path and refuse to expand it. Therefore, we estimate g* and h* with functions g and h and let f = g+h. The function g now represents the best known cost from the start node to node n and h represents a heuristic cost estimate from node n to the goal node. We say that a heuristic function h is admissible if h( n) h * ( n) for n N is satisfied. Therefore an important aspect of our research is a development of good heuristic functions so that h(n)/h*(n) is as close to 1 as possible without going over. A* is an algorithm that performs better with better information. We require an admissible heuristic function for use in A* and therefore we want a lower bound on the time to reach the goal node from any given node. Due to the size of N and of b(n), we have developed successor set generation logic that is implemented in the coding of the A* algorithm. This logic is used to dynamically create the successor set of a node, all connecting arcs, and all costs (arc costs and both functions g and h) as needed. With many shortest path solution techniques, complete knowledge of N and A is required or the algorithm could fail at some point during implementation. However due to the manner in which A* selects the next node for expansion, knowledge of certain nodes and the connecting arcs is only required if these nodes are selected for expansion. Therefore, entire portions of the network that are theoretically included in the network G will never be created during implementation of the algorithm. This intentional exclusion allows A* to save significant amounts of runtime. The logic we have developed is based on the notion of a feasible container move. We consider in which storage areas a specific container may be stored and/or moved to from its current storage slot. We account for the physical restrictions of the storage areas such that a container cannot be stacked higher than the storage areas stack height or outside the other dimensions. The container cannot be placed in a storage slot which does have a container occupying the storage slot immediately below. Finally, we incorporate the path cost logic when dealing with a container undergoing an inspection. As we explore the suite of variants, this logic will change accordingly. As part of the successor set generation logic, we calculate the heuristic function h for each node created. Initially we have utilized a simple method for this calculation. We compare the current state to the goal state, element by element. For each state element that represents a storage slot, if the elements of each state are different, then the current node is not equivalent to the goal node. We therefore include the cost of moving the container from the current storage slot directly to the goal storage slot. We ignore the fact we may not be able to reach the container in the current slot since other container may be stacked on top. For state elements that represent the inspection status, if the elements of each state are different, then the current node is not equivalent to the goal node and we include the cost of an inspection, InspTime. The sum of all these costs will be the heuristic cost at the current node. Better calculation methods for the heuristic function will be developed in the future.

7 7 VI. CONCLUSIONS In this paper, we have described an approach for aiding the management of a container transshipment seaport to decide on the balance between the percentage of containers to undergo security inspection and the concomitant departure delays of outbound vessels. We focused on a highly simplified problem formulation. This problem formulation was intended to be a conceptually and computationally simple starting point on which more realistic problems can be based. We then discussed various ways in which the base case can be realistically extended; many of these extensions are topics for future research. We then used a best-first heuristic search procedure, A*, to model the transshipment seaport container management problem given by the simplified formulation. REFERENCES [1] World Trade Organization, International Trade Statistics 21. Report X World Trade Organization, 21. [2] United States Customs Service, Factsheet: U.S. Customs Container Security Initiative to Safeguard U.S., Global Economy, 22 Archived Press Releases. Feb Accessed March 6, 22. [3] Transportation Research Board, Deterrence, Protection, and Preparation: The New Transportation Security Initiative, Transportation Research Board Special Report 27, TRB, National Research Council, Washington, D.C., 22. [4] American Association of Port Authorities. February 28, 22. WORLD PORT RANKING Accessed July 3, 22. [5] J.E. Lee-Partridge, T.S.H. Teo, and V. K. G. Lim. Information technology management: the case of the Port of Singapore Authority., Journal of Strategic Information Systems, vol. 9, 2, pp [6] M. Kia, E. Shayan, and F. Ghotb, The importance of information technology in port terminal operations, International Journal of Physical Distribution & Logistics Management, vol. 3, No. 3/4, 2, pp [7] E. Kozan, Optimising Container Transfers at Multimodal Terminals., Mathematical and Computer Modelling, vol. 31, 2, pp [8] P. Preston and E. Kozan, An approach to determine storage locations of containers at seaport terminals, Computers & Operations Research, vol. 28, no.1, 21, pp [9] A. Narasimhan and U. S. Palekar, Analysis and algorithms for the transtainer routing problem in container port operations, Transportation Science, vol. 36, 22, pp [1] K. H.Kim and K. Y. Kim, An optimal routing algorithm for a transfer crane in port container terminals, Transportation Science, vol. 33, 1999, pp [11] J. Pearl, Heuristics: intelligent search strategies for computer problem solving. Addison-Wesley Publishing Company, Inc., Reading, MA., 1984.