INTERBLOCK CRANE SCHEDULING AT CONTAINER TERMINALS Omor Sharif University of South Carolina Department of Civil and Environmental Engineering 00 Main Street Columbia, SC 0 Telephone: (0) -0 Fax: (0) -00 Email: omor.sharif@gmail.com Length of paper: Text = 00 words, Number of Figures = ; Number of Tables = ; Total = 0 words Submission Date: April, 0 0 0
ABSTRACT It has been recognized that efficiency in yard operations are critical for the overall productivity of the container terminal. Yard cranes are popular and frequently used equipment for handling, reshuffling and managing flow of containers in a yard. An efficient deployment plan for these cranes is required in response to the varying workload among yard blocks in different planning periods. This project studies an algorithm and mathematical model proposed in literature to address the crane deployment problem. The crane deployment problem is, given the forecasted workload of each block in each period of a day, assign the crane among blocks dynamically so that the total unfinished workload in the yard is minimized. Keywords: Rubber tired gantry crane, container terminal, scheduling, yard operations. 0
Sharif 0 0 0. INTRODUCTION AND PROBLEM STATEMENT The globalization of trade and subsequent growth of containerization for transporting goods in containers have brought many difficulties and challenges in marine terminal operations. Capacity constraints, lack of adequate decision making tools, congestion and environmental concerns are some of the major issues faced by the container terminals today. Increasing containerization has also resulted in increased complexity in planning for terminal managers to provide satisfactory customer service and maintain terminals competitiveness. Various operations research optimization techniques, automated equipment and information technology have become indispensible for efficient management of marine terminal operations and to attain high productivity in container flow with limited resources. Marine terminal operations involve various logistics processes and deployment of expensive resources. Thus efficient decision making is imperative in each process to obtain optimum results. This project addresses an important planning problem in container yard operation. It has been recognized that efficiency in yard operations are critical for the overall productivity of the terminal. The efficiency and quality of management is the container yard operations affect all terminal decisions, related to the allocation of available handling equipment and the scheduling of all activities (Rashidi and Tsang, 00). The storage yard at a container terminal serves as a temporarily storing area for containers before they are picked up by a truck from the land side or they are ready to be loaded into a vessel. Since large terminals have a sizable land space for storage, the yard is typically divided in multiple yard zone and each yard zone is further divided into multiple rectangular shaped yard blocks. Each block contains several lanes of space for storing containers in stack, movement of cranes and parking of transport vehicles for pickup/delivery operations. Figure shows a sample layout of a container terminal with two yard zone and six yard blocks in each yard zone. One very popular equipment used for loading, unloading, rehandling/reshuffling of containers in container yard is Rubber Tired Gantry Crane (RTGC). RTGCs are popular and more frequently used in large terminals with high container flows and other automated technologies (Henesey, 00). An RTGC travels on rubber tired wheels spanning over a yard block width. Figure shows an RTGC standing over a lane yard block. Compared to Rail Mounted Gantry Cranes (RMGCs) which can only travel in one direction across the stacks, RTGCs offer more flexibility since they can be transferred to different blocks as required. Since the workload among the blocks varies throughout the day, the deployment of RTGCs is an important decision making problem the terminal managers have to consider. An efficient deployment plan is necessary to address variable workload demand and is imperative for overall productivity of yard operations and other related processes in a container terminal. This project studies the algorithm and mathematical model proposed by Linn et al. (00) to address the crane deployment problem. The crane deployment problem is, given the forecasted workload of each block in each period of a day, assign the crane among blocks dynamically so that the total unfinished workload in the yard is minimized.
Sharif FIGURE A layout of a container terminal (Linn et al, 00). FIGURE RTGC moves on rubber tired wheels spanning over seven lanes of space or a block space. Each block has seven lanes of spaces: six lanes for container storage and one lane for tracks (Linn et al, 00).
Sharif 0 0 0. BACKGROUND AND LITERATURE REVIEW There is a vast amount of literature in the area of marine container terminal modeling. With container terminal operations becoming more and more important, an increasing number of publications on container terminals have appeared in the literature. A comprehensive review of previous work is beyond the scope of this paper. For a comprehensive survey of container terminals related research, see Vis and Koster (00), Steenken et al. (00), Stahlbock and Vob (00), Crainic and Kim (00), Murty et al. (00), Rashidi and Tsang (00), Vacca et al. (00), Henesey (00). The following review is limited to published works that pertain to yard crane scheduling at seaport container terminals. Also, it should be noted that, in general, there are two types of cranes deployed at a container terminal; namely yard cranes and quay cranes, both of which have been studied extensively. In subsequent discussions, the term crane refers to yard cranes unless stated otherwise. Zhang et al. (00) addressed the dynamic crane deployment problem where given the forecasted workload of yard blocks in each period of a day, the objective is to find the times and routes of crane movements among yard blocks so that the total delayed workload in the yard is minimized. A mixed integer program (MIP) was developed and solved using Lagrangean relaxation. Interblock crane deployment has also been studied by Cheung et al. (00), Linn et al. (00) and He et al. (0). However, these studies do not stipulate detailed work flow for the cranes in serving the trucks. Kim et al. (00b) studied various truck serving rules using simulation to minimize truck delay. The sequencing rules comprise dynamic programming, first-come-first-served, unidirectional travel, nearest-truck-first-served, shortest-processing time rule, and a rule set from reinforcement learning. Ng and Mak (00) studied the problem of scheduling a yard crane to handle a given set of jobs with different ready times. They proposed a branch and bound algorithm to solve an MIP that finds an optimal schedule that minimizes the sum of truck waiting times. In a follow-up study by Ng (00), the author extended his previous work to deal with multiple yard cranes instead of a single yard crane. His model accounted for interference among cranes which may occur when they are sharing a single bi-directional traveling lane. An integer program was proposed and a heuristic was developed to solve the model. Although this work focused on the yard crane scheduling problem to expedite vessel operations, the proposed model and solution methodology are applicable to drayage operations. In contrast to inter-block deployment studies the study provides detail schedule for handling of individual containers. Lee et al. (00) studied the scheduling of a two yard crane system which serves the loading operations of one quay crane at two different container blocks, so as to minimize the total loading time at stack area. A simulated annealing algorithm was developed to solve the proposed mathematical model. Li et al. (00) developed a crane scheduling model where operational constraints such as fixed yard crane separation distances and simultaneous container storage/retrievals are considered. The model was solved using heuristics and a rolling-horizon algorithm. Most of the existing studies that address the yard crane scheduling problem have taken the centralized approach which employs mathematical programming models such as integer programs or mixed integer programs. On the other hand a very few studies employed decentralized approaches such as agent-based modeling which is a relatively new research field within the realm of artificial intelligence. Huynh and Vidal (0) and Vidal and Huynh (0) introduced an agent-based approach to schedule yard cranes with a specific focus on assessing
Sharif the impact of different crane service strategies on drayage operations. In their work, they modeled the cranes as utility maximizing agents and developed a set of utility functions to determine the order in which individual containers are handled. 0. METHODOLOGY The container yard operational hours in this study is assumed to be composed of three shifts a day. Each shift can be further divided into two planning periods each of hours length to better manage the varying workload throughout the day. Thus in this study a day is partitioned into six planning periods: 00:00 :00, :00 :00, :00 :00, :00 :00, :00 0:00, and 0:00 :00. Typically, at each day an estimate of the workload for the next day at each block of the yard is available. For safety considerations, a maximum of two RTGCs are deployed per block to handle containers. Though, it is possible to move the RTGCs among different blocks as required, frequent transfers may lead to traffic congestion, the maximum number of interblock movement of an RTGC is kept limited to once each period in this implementation. Two types of RTGC movements can be distinguished based on the time required for the movement. The first type of movement is between adjacent blocks where an RTGC does not need to turn its wheels because origin and destination blocks share the same traveling lane. The second type of movement is between parallel blocks where an RTGC needs two 0 degree turn of wheels because the origin and destination blocks have different travel lanes. The latter type of movements requires additional time. We assume that first type of movement takes minutes and second type of movement takes an extra minutes ie minutes. For instance, in Figure, to transfer an RTGC from block W to W, minutes transfer time is assumed. If an RTGC is transferred from block W to W or W, minutes of transfer time is assumed. 0 FIGURE RTGC transfer time between two blocks (Linn et al, 00). If the expected number of moves to be performed in each block is known, the expected workload can calculated using the following formula in term of crane time units. Workload (crane time units) in a block = Number of moves in the block x Average time required for each move The objective of RTGC deployment is to minimize the remaining workload (overflow) from one period to the next. To simplify the deployment problem, three types of blocks can be distinguished. They are sink blocks, source blocks and neither blocks.
Sharif 0 0 0 a) Sink blocks: Sink blocks meet the following criteria- Workload > Crane Capacity and Number of Cranes available < Thus sink block needs and can take additional RTGCs. b) Source blocks: Source blocks meet the following criteria- Workload < Crane Capacity and Number of Cranes available Thus source block have extra capacity and can take spare RTGCs. c) Neither blocks: Neither blocks meet the following criteria- Workload = Crane Capacity or Workload > Crane Capacity and Number of Cranes available = Since neither blocks do not need or cannot receive RTGCs, they are discarded from analysis to save computation time. RTGCs can be transferred from source blocks to sink blocks after completion of work in their current block as to spend the extra capacity in sink blocks. However, since this transfer involves travel time between blocks, extra crane minutes available must be greater than travel time, otherwise a transfer is not allowed. Some parameters are defined first for algorithmw i The total workload of block i in the beginning of a period. r i The number of RTGCs in block i at the beginning of the period. T c The total crane-minutes/crane/period. T c =0 for a -hour planning period. N c the maximum number of cranes allowed in each block at any time. N c = for the current case. The deployment algorithm is implemented in two steps. They are ) presort step and ) RTGC deployment step.. Presort step identifies eligible RTGC and sink blocks: For i =, n. If w i T c r i and r i = N c ; remove block i and its RTGCs from further consideration.. If w i = T c r i, remove block i and its RTGCs from further consideration.. If w i > T c r i and r i < N c, make block i a sink block with unfulfilled workload, w i = w i - T c r i. Its RTGC needs to stay in the block for the entire period and is therefore, removed from further consideration.. If w i < T c r i ; then remove the block from further consideration, but it has v i = T c r i - w i excessive RTGC-minutes available. Set n a = Integer(v i /T c ) +.Make n a RTGCs in the block available. If (v i /T c ) ; make one of the RTGCs in the block eligible with v i minutes (i.e. if this RTGC is selected for transferring to new block j, it will begin its transferring immediately after completing w i = T c r i - v i minutes in block i). If < (v i /T c ) ; make two RTGCs currently in the block eligible. One of them can be transferred after completing o i minutes work, and the other can go after completing p i minutes of work in the block i; where o i + p i = w i and o i 0 and p i 0:
Sharif If < (v i /T c ) ; make three RTGCs in the block eligible. One can be transferred after completing o i minutes work; the second crane can go after completing p i minutes of work in the block i; the third crane can go after q i minutes where o i + p i + q i = w i and o i, p i, q i 0, etc.. RTGC deployment step identifies the optimal deployment plan for the source and sink blocks. Let: N number of eligible RTGCs. M number of sink blocks. u + i amount of work overflow in block i to next period. u - i amount of work underflow in block i to next period. w i unfulfilled workload (in RTGC-minutes) at sink block i: v j crane-minutes available from eligible RTGC j: S j set of sink blocks to which eligible RTGC j can be transferred. T i set of all eligible RTGCs, which can move to sink block i: c ij travelling time in minutes to go from the block where RTGC j is to block i: γ i = if sink block i has one RTGC stationed in it currently; = if sink block i has zero RTGC stationed. The mathematical program is presented below- 0
Sharif 0 In the optimum solution of this model, the y ij provides the optimum deployment of the eligible cranes that minimizes the total amount of work overflow to next period. Constraints () are to insure that one crane will be moved to only one sink block. Constraints () insure that cranecapacity brought into a sink node would be same as the additional crane-capacity needed. Constraints () are to insure that there will be not more than two cranes in each block. Constraints () are nonnegativity constraints and Constraints () are the binary constraints. For the sink blocks i = to M; u i is the amount of work overflow to the next period. These values are to update the workloads of each block for the next period before repeating the deployment algorithm for the following period.. IMPLEMENTATION AND RESULTS The presorting step of the crane deployment algorithm was carried out in Microsoft Excel. Excel is suitable to organize data such as system s initial condition, layout of blocks, workload at each block at different planning period, number of RTGCs initially deployed and transfer time of RTGCs etc. The optimal RTGC deployment step, which is a mixed linear integer program, was implemented in CPLEX, an optimization programming environment developed by IBM ILOG Corp. The solution to a small problem by applying the crane deployment algorithm will now be presented. The sample problem has four yard blocks and their layout is as shown in Figure. BLOCK ( RTGCs) BLOCK (0 RTGCs) BLOCK ( RTGCs) BLOCK (0 RTGCs) FIGURE Block layout for the sample small problem
Sharif TABLE Data for the sample problem Block Workload Initial Crane ID (minutes) RTGCs Capacity Note Eligible RTGCs Unfulfilled Workload Extra Crane Capacity w i T c *R i w i ' v j 0 0 Source 0 0 Sink 0 Source 0 0 0 Sink Total 00 0 Capacity> Workload Table shows the workload data and initial RTGC assignments at the beginning of a -hr planning period for the four blocks in the studied problem. In fact, the yard operation planner can establish the initial number of RTGCs based on discretion. The only constraint to be ensured is that total number of assigned RTGCs cannot exceed the total available RTGCs and not more than two RTGCs can be assigned per block. The number of RTGCs in each block at the end of the planning period can be treated as the initial number of RTGCs at the beginning of next planning period and the deployment algorithm can be repeated until all six periods in a day is covered. For the sample problem, Table identifies the source and sink blocks and number of eligible RTGCs from each source block. The unfulfilled workload for each sink block and extra crane capacity available from the eligible RTGCs are also shown. Also the transfer times of eligible cranes to sink blocks (c ij ) are shown in Table. TABLE RTGC transfer time for the sample problem Transfer Time Minutes c ij c c c c TABLE Optimal solution for the sample problem Parameter Value Objective u + u + 0 u - 0 u - y, y We assume that the eligible RTGC from block is j= and the eligible RTGC from block is j=. Also, the sink block is i= and sink block is i=. At this point, presort step is done and the data is available to be fed into optimal deployment model. The optimal solution is shown in
Sharif 0 0 0 Table. The crane from block is moved to block and the crane from block is moved to block after finishing their respective share of workload in their origin block. The total overflow of workload at the end of planning period is minutes. The crane experiences as minutes of idle time at block. The algorithm appears to be very efficient in computational time.. CONCLUSION Yard cranes are popular and frequently used equipment for handling, reshuffling and managing flow of containers in a yard. An efficient deployment plan for these cranes is required in response to the varying workload among yard blocks in different planning periods. This project studies an algorithm and mathematical model proposed in literature to address the crane deployment problem. The deployment algorithm was tested on a sample small sized problem and the results were shown. An efficient reallocation of RTGCs to other blocks when no more workload exists in their block can be achieved using the developed mixed integer program. However, the proposed model can be improved in several ways for performance. For example, one RTGC can be moved out earlier than the other RTGC without sharing workload all the way (when two RTGCs exist in a source block). Also, a RTGC can be relocated more than once in a planning period. Furthermore, the algorithm considers one planning period at some time whereas considering multiple planning periods together may offer better RTGC assignment strategies. Also, the detail workflow for each container move within a block is not addressed in this study. REFERENCES Cheung, Raymond K., Chung-Lun Li, and Wuqin Lin. "Interblock Crane Deployment in Container Terminals." TRANSPORTATION SCIENCE, no. (00): -. Crainic, Teodor Gabriel, and Kap Hwan Kim. "Chapter Intermodal Transportation." In Handbooks in Operations Research and Management Science, edited by Barnhart Cynthia and Laporte Gilbert, -: Elsevier, 00. He, Junliang, Daofang Chang, Weijian Mi, and Wei Yan. "A Hybrid Parallel Genetic Algorithm for Yard Crane Scheduling." Transportation Research Part E: Logistics and Transportation Review, no. (0): -. Henesey, L.E. "Multi-Agent Systems for Container Terminal Management." (00). Huynh, N. " An Agent-Based Approach to Modeling Yard Cranes at Seaport Container Terminals." Paper presented at the Proceedings of the Symposium on Theory of Modeling and Simulation., 0. Kim, Kap Hwan, Keung Mo Lee, and Hark Hwang. "Sequencing Delivery and Receiving Operations for Yard Cranes in Port Container Terminals." International Journal of Production Economics, no. (00): -.
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