VEHICLE DISPATCHING PROBLEM AT THE CONTAINER TERMINAL WITH TANDEM LIFT QUAY CRANES. A Dissertation YAO XING

Transcription

1 VEHICLE DISPATCHING PROBLEM AT THE CONTAINER TERMINAL WITH TANDEM LIFT QUAY CRANES A Diertation by YAO XING Submitted to the Office of Graduate Studie of Texa A&M Univerity in partial fulfillment of the requirement for the degree of DOCTOR OF PHILOSOPHY Chair of Committee, Committee Member, Head of Department, Luca Quadrifoglio Bruce Wang Yunlong Zhang Wilbert E. Wilhelm John Niedzweci Augut 2013 Major Subject: Civil Engineering Copyright 2013 Yao Xing

2 ABSTRACT The mot important iue at a container terminal i to minimize the hip turnaround time which i determined by the productivitie of quay crane (QC). The tandem lift quay crane have 33% higher productivitie than ingle lift QC. However, the tandem lift operation bring new challenge to the vehicle dipatching at terminal and thi ha become a big iue in the application of tandem lift QC. The vehicle dipatching at terminal i to enhance the QC productivitie by coordinating the QC operation chedule and the vehicle delivery chedule. The tatic verion of the problem can be formulated a an MILP model and it i a combinational optimization problem. When the type of QC i tandem lift, the problem become more complicated becaue it require two vehicle ide by ide under the QC. Thu, the alignment of vehicle have to be conidered by coordinating the delivery chedule between vehicle. On the other hand, becaue the container are operated alone by the yard crane, the vehicle could not be grouped and dipatched in pair all the time. Thi diertation invetigate the tatic and dynamic verion of the problem and propoe heuritic method to olve them. For the tatic verion, Local Sequence Cut (LSC) Algorithm i propoed to tighten the earch pace by eliminating thoe feaible but undeirable delivery equence. The time window within which the container hould be delivered are etimated through olving ub-problem iteratively. Numerical ii

3 experiment how the capability of the LSC algorithm to find competitive olution in ubtantially reduced CPU time. To deal with the dynamic and tochatic woring environment at the terminal, the diertation propoe an on-line dipatching rule to mae real-time dipatching deciion without any information of future event. Compared with the longet idle vehicle rule, the propoed priority rule horten the maepan by 18% and increae the QC average productivitie by 15%. The enitivity analyi tated that the uperiority of the priority rule i more evident when the availability of vehicle i not ufficient compared with the frequency of releaing tranportation requet. iii

4 ACKNOWLEDGEMENTS I would lie to pecially than my advior and committee chair, Dr. Luca Quadrifoglio for hi patient guidance and mentorhip during the pat five year. I would alo lie to than my committee member, Dr. Zhang, Dr. Wang, and Dr. Wilhelm, for their guidance and uggetion that help me finih thi diertation. Than to UTCM who provided financial upport for my PhD tudy and reearch in Texa A&M Univerity. I alo appreciate Mr. Shi in Nanha Container Terminal, who anwered me quetion about terminal operation and provided me preciou operation data for my reearch. I would alo lie to than Dr. Alan Zhang in Moffatt and Nichol Engineer for hi help and guidance for my reearch. Than alo go to all my friend and clamate at Texa A&M Univerity, epecially thoe in the diviion of tranportation engineering, for their friendhip and help. Finally, I epecially than my deeply loved parent and my huband for their love, patience and encouragement. iv

5 TABLE OF CONTENTS Page ABSTRACT...ii ACKNOWLEDGEMENTS... iv TABLE OF CONTENTS... v LIST OF FIGURES...vii LIST OF TABLES... viii 1. INTRODUCTION LITERATURE REVIEW Terminal Logitic Tranportation Optimization at Terminal Mathematical Algorithm Exact Method Heuritic Algorithm Simulation NANSHA CONTAINER TERMINAL Layout Equipment Container Flow PROBLEM STATEMENT Problem Decription Model Formulation LOCAL SEQUENCE CUT ALGORITHM Motivation Baic Scheme of Local Sequence Cut Algorithm Initial Solution Time Window and Cut-off Delivery Sequence Sub-problem Analyi of the Sub-problem Reult v

6 5.2.5 Proce of Baic Scheme Enhanced Scheme Treatment at Early Stage Size Limit on Sub-problem Overall Proce of LSC Algorithm Numerical Experiment LSC v.. Optimality Time Window and Cut-off Delivery Sequence MODIFIED LSC ALGORITHM Priority Baed Heuritic Method Creation of Aignment Priority Meaurement Selection Mechanim Numerical Experiment Reult Senitivity Analyi Initial Solution Time Limit for Solving Sub-problem ON-LINE DISPATCHING POLICY Priority On-line Dipatching Rule Numerical Experiment Deign of the Experiment Experimental Scenario Numerical Experiment Reult Senitivity Analyi Cycle Time Vehicle Speed CONCLUSION AND FUTURE RESEARCH REFERENCES vi

7 LIST OF FIGURES Page Figure 1 Container flow in the loading and unloading operation... 1 Figure 2 A torage bloc erved by a yard crane (RTGC)... 3 Figure 3 Different type of QC Figure 4 Geographic location of NCT Figure 5 Satellite view and the layout of NCT Figure 6 Part of a quay crane woring equence Figure 7 RTG at the torage bloc Figure 8 Single loop layout at the quay ide Figure 9 Initial delivery equence of the two vehicle in a group Figure 10 Determination of cut-off delivery equence Figure 11 Contruction of ub-problem Figure 12 Number of cut-off delivery equence through iteration Figure 13 Update of time bound through iteration Figure 14 Current dipatching rule Figure 15 Performance under different degree of tochaticity Figure 16 Performance of dipatching rule againt the varied cycle time Figure 17 Performance of dipatching policie againt the varied vehicle peed vii

8 LIST OF TABLES Page Table 1. Empirical Ditribution of QC Cycle Time Table 2. Relationhip Between the Operation, Requet and Ta Table 3. Calculation of Setup Time Table 4. Extra Contraint Impoed to a Sub-Problem Table 5. Analyi of an Upper Bound Sub-problem Table 6. Analyi of a Lower Bound Sub-problem Table 7. Proce of the Baic Scheme for Local Sequence Cut Algorithm Table 8. Entire Proce of Local Sequence Cut Algorithm Table 9. LSC v.. Optimality (Small Size Problem) Table 10. LSC v.. Optimality (Large Size Problem) Table 11. Priority Meaurement in the Firt Step Table 12. Priority Meaurement in the Second Step Table 13. Proce of Modified LSC Algorithm Table 14. Numerical Experiment Reult of Different Method Table 15. Modified LSC Algorithm v.. LB Table 16. Algorithm Senitivity to the Initial Solution Table 17. Algorithm Senitivity to the Time Limit in Solving Sub-problem Table 18. Priority Dipatching Rule viii

9 1. INTRODUCTION The firt regular ea container ervice began in 1961 between and port in the Caribbean, Central and South America. The year 2010 witneed an increae of global containerized trade from around 30 million TEU (twenty feet equivalent unit) in 1990 to 150 million TEU or more in 2010 (76). Not urpriingly, uch dramatic increae impoe the heavy preure on the operation at container terminal. FIGURE 1 Container flow in the loading and unloading operation (61). For a terminal, the mot critical iue i to minimize the hip turnaround time by accelerating operation and moothing the container flow (FIGURE 1). There are two type of operation loading and unloading at a terminal. Upon a hip arrive at the terminal, it i berthed at the quay ide according to the berth allocation deciion and a certain number of quay crane (QC), dependent on the ize of the hip and the number of container, are aigned to load container onto the hip and unload them off the doc 1

10 and hold. When a QC i unloading a container from the hip, it hoit the container and lift it from the hip then poition it on a vehicle for delivery. A fleet of vehicle with unit capacitie, namely truc, automated guide vehicle (AGV), and manned or automatic addle carrier (ASC), tranport container from the quay ide to the torage area. The torage area (yard area) i an open zone ued to tore the import and export container temporarily. It conit of myriad bloc, further differentiated into lane, row (bay) and tac (tier), dependent on the pan and height of the yard crane (FIGURE 2). In each bloc, there are alway one to two yard crane (YC) that tore and retrieve container to and from the bloc. When the vehicle deliver the container to the torage area, the yard crane tac it on it detination torage poition according to the pre-defined torage planning. The loading operation i conducted in the revere order. Apparently, the hip turnaround time i directly determined by the woring peed of QC. In order to further accelerate the operation, terminal in the Netherland, Shanghai, Shenzhen, Dalian, Singapore, Dubai, and other area either have already been equipped with or have conidered inveting in tandem lift QC in recent year (81, 83). A tandem lift QC i deigned to increae productivity by lifting two 40ft container or four 20ft container each time, while the conventional ingle trolley quay crane lift only one 40ft or two 20ft container each time (FIGURE 3). Tandem lift quay crane are equipped with a pecific twin preader. Thi pecial deign facilitate it adjutment to different container height, a well a to ide-to-ide clearance of two container when landing them onto adjacent vehicle. To mae a full ue of tandem lift QC capacity in 2

11 practice, two vehicle are required to arrive at a QC, not necearily at the ame time, However, they hould get ready before a tandem lift, a thee container need to be lifted or releaed together by the QC (54, 83). Tandem lift QC are deigned to accelerate the unloading/loading operation to meet the demand of erving mega veel in a hort time. According to publihed data, the tandem lift QC can provide 33% greater productivity than the traditional ingle trolley QC (64). FIGURE 2 A torage bloc erved by a yard crane (RTGC) (86). However, tandem lift operation poe new challenge and difficultie for the vehicle dipatching problem at terminal. At the operational level, the vehicle dipatching problem concern cheduling a fleet of vehicle to tranport container to enhance productivity of QC and to mooth the container flow. Due to the lac of 3

12 buffer area in mot terminal, the quay crane unload/load a container directly on/from a vehicle. Thu, the QC cannot proceed to it next operation if there i no vehicle waiting under it. According to tudie, the QC actual productivity in operation only achieve around 40-60% of it technical maximum capacity and it waiting for a vehicle availability i one reaon for thi (51, 73). It i well undertood that the QC operation efficiency and the terminal throughput critically depend on the vehicle dipatching operation (34, 28), the main focu in thi diertation. When the quay crane i a traditional ingle lift, the problem involve a combinational optimization that can be categorized in the Vehicle Routing Problem (VRP) (7, 29) or Parallel Machine Scheduling Problem. Converely, when the quay crane are tandem lift, the problem become more complicated becaue of the alignment of vehicle dipatched to pic up or drop off container at the QC (9). To accommodate tandem lift operation, there are uppoed to be two vehicle ide by ide waiting under a QC. Otherwie, the QC and the vehicle that arrive earlier mut wait for other vehicle arrival. Such idling not only delay the QC operation However, alo affect the productivity of vehicle. On the other hand, when the two vehicle leave the tandem lift QC and travel to the torage area, they eparate and become independent again. Becaue the two container operated in the ame tandem lift may be tored in totally different torage bloc and the YC can only operate one 40ft or two 20ft container each time. Thu, we could not imply bind two vehicle a a group and dipatch them together all the time. 4

13 (a) Conventional ingle-trolley quay crane (b) Tandem lift QC FIGURE 3 Different type of QC. Actually, the tranportation of container between the QC and YC ha become a big iue of the application of tandem lift QC. Without an effective operation technique and efficient vehicle dipatching policy, the application of tandem lift QC 5

14 may not increae the productivity a expected. In turn, the lac of uch a policy reduce the productivity of vehicle (1, 64 and 82). In uch a cae, in addition to the coordination between the QC operation chedule and the vehicle delivery chedule, the delivery chedule of different vehicle have to be ynchronized imultaneouly. Nonethele, until now, there exit very little literature that invetigate the pecial woring pattern of tandem lift QC and their influence on the vehicle dipatching problem at the container terminal. No reearcher ha yet olved the problem through employment of mathematical model or algorithm. Thu, the contribution of thi tudy i a follow: Thi tudy conider the woring characteritic of tandem lift QC in the modeling of and olution to the vehicle dipatching problem. In addition, both the model and the propoed method in thi diertation can be applied to the imilar vehicle dipatching problem with precedence relationhip. For the tatic verion of the problem, we formulated the problem a a mixed integer linear program with the objective of minimizing the maepan, the total time it tae to load and unload all container. To olve uch a combinatorial optimization problem, thi diertation propoe an innovative heuritic method to tighten the earch pace by eliminating thoe feaible but undeirable delivery equence. Thi method i capable of finding competitive olution within a ignificantly horter CPU time. For the dynamic verion of the problem, thi diertation develop an on-line dipatching policy to mae deciion without information of future event. The numerical experiment reveal that the propoed method perform better than the 6

15 currently employed dipatching rule. It uperiority i more ubtantial when there are too few vehicle reource compared to tranportation requet. The ret of thi diertation i organized a follow: the econd ection review the related wor and literature in the area of terminal operation and vehicle dipatching problem. The third ection introduce the container terminal from which operation data for thi diertation wa gathered. The third ection i followed by the ection of the problem tatement and the mathematical model. The ubequent two ection preent the algorithm for olving the tatic verion of the problem and ae their performance through a erie of numerical experiment. Then, the on-line dipatching rule i introduced and analyzed in the eventh ection. The concluion and dicuion baed on thi tudy comprie the lat ection of thi diertation. 7

16 2. LITERATURE REVIEW Vehicle dipatching problem frequently arie in logitic ytem. Unfortunately, mot publihed reearch doe not apply to a container terminal due to the container terminal pecific operation characteritic. Thi, in turn, require the development of algorithm that conider the pecial characteritic and contraint aociated with the operation at terminal. In the following, we retrict literature review to the reearch related to the following topic: terminal logitic, tranportation optimization in the terminal, and the optimization method of olving the vehicle dipatching problem. The review i not meant to be exhautive, but rather deal with iue and problem imilar to the one dicued in thi diertation. 2.1 Terminal Logitic Operation reearch method have become the main approache to the optimization of operational iue in container terminal. Terminal logitic can be generally claified into the following categorie. Berth allocation: Berth allocation ideally begin an average of two to three wee before the hip arrival. It main objective are to determine the berthing time and poition of hip along the berth (30, 41, 57 and 63), and to minimize the mooring and proce time of the hi (21, 31 49, and 42). Crane aignment and plit: Depending on the ize of the hip, three to five crane operate at one overea veel and one to two crane for a feeder hip. The 8

17 optimization of the crane aignment and cheduling could be formulated a an MIP model, with the objective of minimizing the um of the delay and turnaround time of all hip (4, 12, and 67) and the maximization of the utilization of the crane capacitie (56). The crane aignment wa alo dicued a part of an integrated optimization problem with berth allocation (66, 56). Stowage planning: Stowage planning aign poition within the hip to pecific categorie of container and conider the container attribute, uch a the detination, weight, or container type. The hipping line and the terminal operator uing an off-line optimization method decide planning. The towage planning heavily affect the loading and unloading equence of container, a major factor for determining the woring equence and chedule of crane and horizontal tranporter (23). It objective focu on the minimization of the number of hift during terminal operation (1, 2, and 13) and the maximization of the utilization of the hip torage capacity (17). Storage and tacing policie: Storage and tacing policie attract increaing attention due to the carcity in land area and continuou growth in containerization tranportation. Thee policie are ued to decide the pecific bloc and lot for each container tored in the torage area. The yard rehuffle play the mot important role in the contruction of torage and tacing policie. Mot related tudie were dedicated to the invetigation of the tac configuration and their influence on the number of yard rehuffle (25, 35, 43, and 48). Beide, the utilization of torage area (37, 36, 38 and 75); the turnaround time of the hip (48, 68); the travel ditance between the quay and yard ide (40, 85) are alo conidered when maing deciion. 9

18 A more comprehenive and detailed review about the logitic and operation problem in container terminal wa given in the wor of Steenen et al. (73), Stahlboc et al. (72), Vi and de Koter (78), Günther and Kim (23). 2.2 Tranportation Optimization at Terminal The tranportation in a terminal can be ditinguihed between horizontal and vertical (tacing) tranport. The former one can be further divided into the quay ide tranport and the land ide tranport. The quay ide tranport refer to the container tranportation between the quay crane and the yard crane, carried out by a fleet of truc, trailer, AGV, or traddle carrier. External truc or train outide the terminal carry out land ide tranport. Vertical tranport refer to the tacing tranport carried out by gantry crane, traddle carrier (SC), rubber-tired gantry crane (RTG), and rail-mounted gantry crane (RMG; alo called automated tacing crane (ASC)). Becaue neither tacing tranport nor land ide horizontal tranport i the concern of thi diertation, we only review related paper on the quay ide tranport in the terminal. A the connection between the operation in the quay ide and the yard ide, optimization of the vehicle dipatching proce hould be integrated with other activitie. Bih et al. (6) addreed the vehicle-cheduling-location problem of aigning a torage location to each import container and dipatching vehicle to the container in order to minimize the total time for unloading a veel. To olve uch an NP-hard problem, an aignment problem baed (APB) heuritic algorithm wa preented. The 10

19 effectivene of the propoed heuritic wa analyzed from both wort-cae and computational point of view. Bih (4) developed a mathematical method for the o-called multiple-cranecontrained vehicle cheduling and location problem (MVSL). The problem i to determine the torage location for import container; dipatch vehicle to container and chedule the loading and unloading operation of the crane with the objective of minimizing the turnaround time of hip. He analyzed the complexity of the problem and proved that it wa NP-hard. He propoed a heuritic algorithm and meaured it effectivene according to the wort-cae performance ratio of the heuritic algorithm. Kang et al. (32) preented mathematical model to optimize the number of the quay crane and truc with the objective of minimizing the operation cot. A Marovian-baed deciion model wa propoed a a complement to the queue model to handle generally ditributed ervice time and non-teady-tate operation. However, their tudy only focued on the unloading operation at the container terminal. Koo et al. (44) propoed a fleet management procedure for the tranportation ytem in a terminal. The objective of the management i to imultaneouly find the minimum fleet ize and route for each vehicle while atifying all requirement. The firt phae wa contructed to obtain a lower bound on the fleet ize. Then a tabu earchbaed procedure wa preented for the travel route for each vehicle, atifying all the tranportation requirement within the planning horizon. Meerman and Wagelman (55) conidered an integrated problem that involved the cheduling of different equipment at automated terminal. They preented a branch- 11

20 and-bound algorithm and a heuritic beam earch algorithm to minimize the maepan of their chedule. However, they only conidered loading operation in their wor. Murty et al. (58, 59) decribed and analyzed a variety of interrelated activitie of daily operation at a container terminal and propoed an integrative deciion upport ytem (DSS) for deciion maing. The objective of the DSS i multi-folded and included the minimization of waiting time for external truc, minimization of the congetion inide and outide the terminal, and o on. The propoed DSS wa evaluated at a terminal in Hong Kong and revealed a reduction of 30% in a veel turnaround time, a well a a reduction of 35% in container handling cot. Qiu and Hu (69) preented a bi-directional path layout and an algorithm for routing AGV without conflict and to minimize the pace requirement of the layout. The routing efficiency wa analyzed in the term of the ditance and time AGV need to complete all tranportation job. Saanen et al. (71) compared the productivity of the automated container terminal (ACT) where the AGV or automated lift vehicle (ALV) are ued a the horizontal tranporter. Their performance were compared uing imulation and cot modeling. Vi et al. (79) developed a networ formulation and a minimum flow algorithm to determine the neceary number of AGV required at a emi-automated container terminal. The algorithm propoed wa a trongly polynomial time algorithm and uable in warehoue and manufacturing ytem, a well. 12

21 2.3 Mathematical Algorithm To olve the mathematical model of vehicle dipatching problem and evaluate different vehicle dipatching policie, the exiting approache can be generally divided into two categorie: imulation model and mathematical olution; a well, the latter can be divided further into exact method and heuritic method Exact Method Due to the extremely high computational burden, there ha been very little reearch attempted to acertain exact olution to the vehicle dipatching problem. Langevin et al. (45) propoed a dynamic programming baed method to olve intance with two vehicle. Correa et al. (11) olved the dipatching and conflict-free routing of AGV by combining contraint programming for cheduling and mixed integer programming for conflict-free routing Heuritic Algorithm Although the exact method can guarantee the optimal olution, it unacceptable computation time i the main obtacle to it application. Thu, further effort have been dedicated to propoe an effective heuritic method that olve the problem quicly with tolerable error. 13

22 Ng et al. (62) formulated the vehicle dipatching and cheduling problem in a terminal a an MILP and olved it uing the genetic algorithm (GA) with greedy croover technique. The numerical experiment revealed that the propoed GA method outperformed the GA method that employed other croover cheme. However, their model did not conider the availability of the crane, and each job tart a long a the vehicle arrive. Local earch technique are heuritic that tart from an initial feaible olution and move locally in the neighborhood of the olution pace. The main drawbac i that the olution found might be a local ideal but potentially far from the global bet. Due to the varied technique for ecaping the local optimum, the local earch technique i alo a claic and effective method to olve vehicle dipatching problem. Chen et al. (10) preented an integrated model to chedule the different equipment in a container terminal. They olved the problem uing tabu earch algorithm and developed certain mechanim to aure the quality and efficiency of the algorithm. Homayouni et al. (27) olved the integrated cheduling of quay crane and AGV by uing imulated annealing algorithm. They invetigated the effect of initial temperature and the number of trial on the algorithm and compared them with the reult obtained from olving the MILP model uing branch-and-bound method. In addition to claic earch algorithm, there are plenty of heuritic method deigned to deal with characteritic of the woring environment and the operation requirement. Some of them were propoed a an on-line dipatching method for practical application. 14

23 Hartmann (26) propoed a general optimization model for cheduling job at the container terminal. The model could be applied to AGV, traddle carrier, gantry crane, and worer. He propoed a priority rule baed heuritic method and developed a genetic algorithm for olution. However, they only conidered the loading operation in the reearch, and all the AGV return to a common point after delivering a container. Bih et al. (5) developed an eaily implementable heuritic algorithm baed on the greedy and revered greedy algorithm. They identified both the abolute and aymptotic wort-cae performance ratio of thee heuritic. The above three tudie did not conider the job ready time in the cheduling deciion. Kim and Bae (39) developed a mixed integer programming model for aigning optimal delivery ta to vehicle and propoed a loo-ahead vehicle dipatching method to minimize the total idle time of a quay crane reulting from the late arrival of AGV. Briorn et al. (8) propoed an AGV dipatching trategy baed on inventory management. The imulation tudy revealed the inventory-baed trategy outperformed the conventional due-date-baed trategie. However, their model required buffer area at the quay ide, and they did not conider the precedence relationhip in container dicharging proce. A oppoed to the aumption of fixed AGV capacity in almot all reearch, Grunow et al. (19, 20) propoed a flexible priority rule for dipatching multi-load AGV. Their numerical experiment revealed that the developed pattern-baed off-line heuritic outperformed conventional on-line dipatching rule ued in flexible manufacturing ytem (FMS). 15

24 Lim et al. (50) propoed an auction-baed aignment algorithm in the ene that the dipatching deciion were made through communication among related vehicle and machine for matching multiple ta with multiple vehicle. Their method conidered future event and the performance of the method wa compared through a imulation tudy. Egbelu and Tanchoco (16) invetigated thoe popular AGV dipatching rule in the manufacturing ytem and claified them into two categorie. Vehicle-initiated rule are applied when an idle vehicle appear, and wor-center-initiated rule are applied when there i a delivery ta available. Their tudy uggeted the modified firt come firt erved (MFCFS) rule outperformed other rule. They alo concluded that ditance baed rule are enitive to the guide-path layout and the location of pic-up and dropoff tation. Egbelu (15) poited that the ource driven rule are not uitable in the ytem baed on jut-in-time principle. He propoed a priority dipatching rule baed on the demand tate of the load detination. They uggeted that AGV hould be firt dipatched to delivery ta bound for input buffer whoe length are below a threhold value. Kim et al. (33) uggeted an AGV dipatching method in a manufacturing job hop environment. They propoed a procedure to dipatch AGV for balancing worload among different machine, a well a the worload between the machine and the tranporter. 16

25 Taghaboni-Dutta (74) propoed Vehicle Aignment by Load Utility Evidence (VALUE) method for the aignment problem. The objective of their method attempted to achieve a higher throughput of machine by uing a value-added approach to determine the AGV aignment Simulation Due to the numerou tochatic and uncertainty element, for intance hip delay, equipment failure, human factor, and o on, dicrete event imulation approach i alway ued to evaluate and compare the operation proce and equipment intallation. Simulation reult provide valuable deciion information and upport guidance for the deign of the terminal and the logitic proce (18). Bae et al. (3) compared the performance of AGV and automated lifting vehicle (ALV) in an automated container terminal through imulation experiment and concluded that ALV reached the ame productivity level a the AGV, uing far fewer vehicle. Duineren and Ottje (14) developed a imulation model to determine the number of AGV, maximum AGV peed, crane capacity, and tac capacity. Huo et al. (28) examined the influence of the quantity and the peed of the internal truc on the handling time of hip, uing Witne imulation oftware. Thee reearcher indicated that the quantity of truc hould be configured dynamically, according to the different tage of handling and that increaing peed of truc would not necearily improve the handling efficiency of the crane. 17

26 Lau et al. (52) dicued the relationhip between the number of AGV and the terminal layout via imulation and concluded that the yard layout ha inevitable effect on the terminal performance, a well a on the number of AGV. Liu and Ioannou (53) compared four different vehicle-initiated AGV dipatching rule in an ACT from apect of the throughput of quay crane, average waiting rate of AGV in the quay crane queue, and o on. Par et al. (65) developed a imulation model to compare the performance of different tranporter in an ACT, uing ARENA pacage. Baed on the imulation reult, they concluded that automated tacing crane i more efficient than AGV. Vi et al. (80) analyzed the minimum vehicle fleet ize under time window contraint at a container terminal. In their tudy, there were buffer area under the quay crane and in the torage area. The tranportation of container mut tart in the time window, determined in advance. Yang et al. (82) and Vi and Haria (77) examined and compared the effect of uing AGV and ALV on unloading time of a hip through imulation tudie. Yun and Choi (84) developed an object-oriented approach to imulate the operation of a container terminal uing SIMPLE++ imulation oftware. 18

27 3. NANSHA CONTAINER TERMINAL The operation data ued in thi tudy i provided by Guangzhou Nanha Container Terminal Phae I (NCT) in Guangzhou, China. NCT i located in the outheat of Guangzhou City and at the etuary of Pearl River, the geographic and geometrical center of Pearl River Delta, a een in FIGURE 4. The buinee at NCT include container loading/dicharging, tacing, warehouing, manufacturing, proceing, maintenance, and o on. FIGURE 4 Geographic location of NCT. 19

28 3.1 Layout The terminal i 1400m long and 1300m wide with a water depth of 15.5m. The layout of NCT i a typical parallel, ide-loaded terminal layout (FIGURE 5) where the container tac are parallel with the berth. Thi type of layout i the mot popular in Aian container terminal. All the QC are equipped at the wateride of the terminal, and all the torage bloc are located at the other ide. Rubber-tired gantry crane (RTGC) and rail-mounted gantry crane (RMGC) are alway ued a yard crane in the torage area (46, 47). Vehicle can par at a ide of a bloc where the container i piced up (put down onto) the vehicle. The path area in the quay ide i a multi-lane loop layout where vehicle travel in a ingle loop. It i very convenient and imple for traffic control at the terminal. The loop i a fixed equence of picing up and dropping off point at QC and YC. FIGURE 5 Satellite view and the layout of NCT. 20

29 3.2 Equipment The quay ide of NCT i equipped with ixteen quay crane. Thirteen are conventional ingle trolley QC, and three are tandem lift QC. A few day prior to the arrival of a hip, the terminal will be informed of the tatu of the hip and container. Then the terminal operator mae the plan of quay crane allocation and generate woring equence for thoe QC and the torage plan of import container, according to the rule and regulation in the terminal. The generation of the woring equence ha to conider many element, lie the weight, ize, and content of the container, the contract between the hipper and the terminal, the tability of the hip, and o on. FIGURE 6 i part of a QC woring equence, generated uing the oftware in NCT. The content of the woring equence i explained in the following way: SEQ the number of the operation equence; D/L the type of operation, where D repreent dicharge (unload) and L repreent load the container; F/E the weight of the container, where F repreent that the container i full and E a empty; CNTR No. Each container i labeled with a unique code number. Thi code number i ued for the operator to trac the information and tatu of every container. During the operation proce, the operator and the manager can acce the information of a container by canning it code number. FROM POS. and TO POS. the container torage poition on the hip. In the unloading proce, the QC unload the container from the hip and the TO POS. 21

30 column i empty. During the loading proce, the quay crane load the container onto the hip, and the FROM POS. column i empty. FIGURE 6 Part of a quay crane woring equence. Unle encountering an unexpected event, the container are loaded/unloaded following the QC woring equence, predetermined according to the towage planning and other detailed information ent to the terminal, uch a the hitorical average cycle time and travel time (23, 60). The cycle time of a QC i the time required to tranfer a container between the hip and a vehicle. The cycle time depend, for example, on the deign capabilitie of the crane, the operation experience of the crane 22

31 driver, the type of hip, and the poition of the container in the hip. The cycle time of the tandem lift QC ued in thi tudy i baed on the record of 10hr operation in NCT. According to the empirical ditribution lited in TABLE 1, the time interval econd tae the highet percentage. The cycle time ued in the mathematical model i uppoed to be the QC deigned woring peed, auming there i no idling due to other external influence. Thu, the cycle time interval horter than 50 econd or longer than 140 econd are not included. Thee long cycle time might due to the mitae record or poible large diturbance, lie breadown of equipment, wrong information on container, or generally unexpected event and mitae. TABLE 1 Empirical Ditribution of QC Cycle Time Cycle time (ec) Fraction The yard crane employed in the torage area of NCT are rubber-tired gantry crane (RTG, FIGURE 7). Among the even lane under each RTG, ix lane are ued to toc the container and the other one i ued a the paenger lane for truc paing. 23

32 The height of the container taced in the torage area i alway fiver layer. There are total eighty bloc in the yard area for the container temporal torage and 46 RTG for toring and retrieving container. The RTG can travel between the bloc according to the woring load among different bloc. There i alway one or two RTG tac and retrieve container at each bloc. FIGURE 7 RTG at the torage bloc. 3.3 Container Flow To facilitate undertanding the container flow in the terminal and interrelated activitie of different equipment, we preciely decribe the unloading proce a an example. When a tandem lift QC tart unloading a pair of container following it woring equence, two truc are choen from the pool of idle vehicle to pic up the two container at the QC ide. When thee two vehicle have arrived there and top under the QC ide by ide, the QC tart to poition the container on them; otherwie, 24

33 the QC ha to wait for them, delaying and dirupting the current unloading operation and the ucceeding operation in that QC woring equence. In contrat, the two vehicle might arrive before the QC i ready to poition the two container onto them. Obviouly thi cae doe not generate any additional waiting time for the QC. However, it might affect other operation availabilitie of vehicle. After thi tep in the proce, the two vehicle travel to the two container torage bloc and drop them off there. When the vehicle arrive at the torage bloc, the vehicle mut top in front of the bloc and wait for the ignal. When the yard crane i available, the vehicle proceed to the paenger lane and the YC lift the container from the vehicle and tac it to it detination poition according to the command ent from the control center. Note that the torage bloc for the two container in the ame tandem lift might be different from each other. For the loading proce, ince it i conducted in the revere order of unloading proce, we do not elaborate here further. 25

34 4. PROBLEM STATEMENT The problem central to thi diertation i to ee a way to dipatch a fixed number of vehicle to tranport container between the tandem lift QC and YC with the objective of minimizing the maepan or the total time it tae to finih loading and unloading all container in the planning horizon. For the purpoe of modeling, the layout of NCT i implified a a ingle loop layout with multi-lane path without loing the main characteritic (FIGURE 8). The QC are lined up in a row along the berth, while all torage bloc are located at the oppoite ide. The container are tranported by a fleet of truc that travel in a counterclocwie direction. All the vehicle are identical and have the ame unit capacity. Each vehicle can carry only one 40ft container or two 20ft container at a time. In the remainder of thi diertation, two 20ft container are treated a one 40ft container ince they are alway operated and tranported together. The cycle time (denoted by ) of a tandem lift QC i defined a the duration neceary to tranfer two container from a hip (vehicle) to a vehicle (hip). In an unloading proce, the cycle time tart from the moment the QC lift container from a dec or a hold, and end at the moment the QC i ready to poition them onto vehicle. In a loading proce, it tart from the moment the QC lift container from vehicle, and end at the moment the QC poition them onto the hip. 26

35 FIGURE 8 Single loop layout at the quay ide. 4.1 Problem Decription The QC equipped along the berth and the vehicle ued for the tranportation are numbered a q1, q,, 2 q and,, 1, 2 m v v v, repectively. All the operation in the 1 2 q woring equence are numbered a o, o,, o, and each could be loading or unloading operation. The two container operated by q a it th operation are referred to a,1 c and,2 c. The number 1 and 2 in the upercript refer to the two container in the ame tandem lift. When a QC tart one operation, a tranportation requet r and four ta are generated correpondingly. Tae for intance, the following: when o i a loading operation, the tranportation requet r i to tranport the two container,1 c and c,2 from q to their detination bloc where the YC tac them to their detination poition. And two vehicle are required to complete two picup ta (,1,2 p and p ) of container,1 c and,2 c at 27 q and two drop-off ta ( d,1 and d,2 ) of

36 them at the YC. The relationhip between the operation, the tranportation requet and the ta are ummarized in TABLE 2. TABLE 2 Relationhip Between the Operation, Requet and Ta Unload o from the hip Load the hip,1 c and,1 c and c,2,2 c onto,1 Tranport c and QC to the YC Tranport r QC ide ta YC ide ta,1 c and the YC to the QC c,2 c from the,2 from,1,2 p and p,1 d and d,2,1 d and d,2,1,2 p and p Given a number of vehicle and QC with predetermined equence of operation, vehicle dipatching problem i to dipatch vehicle to complete all the ta with the objective of minimizing the maepan. A long a the aignment of vehicle are decided, the chedule of all ta are decided at the ame time. In addition to thoe contraint hared among thoe claical vehicle routing and cheduling problem, there are extra contraint to account for the woring characteritic of tandem lift operation: (1) Precedence Contraint: Since all the container are operated following the pre-defined woring equence, their tranportation could not violate their precedence relationhip. In other word, the container c,1 and c,2 cannot be piced up or dropped off by any vehicle before the container ',1 c or ',2 c if '. Conequently, the earliet tarting time of QC ide ta, pic-up or 28

37 drop-off ta whoe location are at the QC are heavily dependent on the container delivery equence. (2) Simultaneou Contraint: Becaue of the tandem lift operation, the two container in the ame tandem lift hould be piced up or dropped off by the vehicle at the ame time. In other word, the tarting time of their correponding QC ide ta hould be the ame. Thu, the tarting time of a QC ide ta,1 d or,1 p depend on not only the tranportation of c,1, but alo the delivery chedule of,2 c. (3) Capacity Contraint: Becaue of the vehicle unit capacity, the two container in the ame lift cannot be tranported by the ame vehicle. 4.2 Model Formulation Baed on the above decription, the problem can be formulated a an MILP model with the objective of minimizing the maepan. The parameter and the notation that will be ued in the diertation are preented a follow: i, j, t mn, g o the index of ta the index of vehicle the index of quay crane the index of the operation in a QC woring equence the index of the two container operated in one tandem lift the th operation in the woring equence of q 29

38 c the container unloaded/loaded in the operation g, o p the pic-up ta of container g, c g, d g, the drop-off ta of container c g, y g, the quay ide ta of container c ta whoe g,, i.e. the pic-up or drop-off location i at the QC ide. For unloading operation y p, g, g b e C V Q while for loading operation the dummy begin ta the dummy end ta the et of all container the et of all vehicle the et of all quay crane y d, g, g S P D the et of all operation in the woring equence of q the et of all pic-up ta the et of all drop-off ta T the et of all ta T P D Y the et of all quay ide ta i, j the travel time from ta i to ta j M the tandem lift QC woring cycle time the time needed to poition (lift) a container from a crane to (from) a vehicle the vehicle average waiting time at a YC a ufficient large poitive number The deciion variable in the model include: 30

39 x, the binary variable which equal to 1 if ta j i immediately i i j completed after ta i by the ame vehicle; the time when the vehicle arrive at the QC/YC where the ta i i; i the time when the ta i tart; maepan; the time when all ta have been completed With the notation, parameter and variable decribed above, the problem i formulated a the following MILP model. min (1) Subject to: xb, j V (2) j P xie, V (3) i D x 1, t T (4) it, i T b x 1, t T (5) t, j j T e x 1, where Q, S, g 1 or 2, g, g p, d (6) j i i, j i, j M ( x -1), i T b, j T (7) M (1 x ), i T b, j T (8) j i i, j i, j, t Y (9) t t +, t T Y t t 1,, Q,, 1 S g (10) g - g >, where, 1 or 2 (11) y y 31

40 =, where Q S (12),1,2, y y t, t T (13) x { 0, 1}, i, j T. i, j (14) The objective i to minimize the time of completing all pic-up and drop-off ta. Contraint (2) and (3) enure that each vehicle deliver at leat one container by aigning the dummy begin and end ta. V denote the cardinal of the et of vehicle. Contraint (4) and (5) enure that each container i delivered by one and only one vehicle. Contraint (6) enure the vehicle that pic up a container at a QC (YC) mut be the ame one that drop it off at a crane. Contraint (7) and (8) define the arrival time when a vehicle execute a ta. Contraint (9) enure that a quay ide ta tarting time cannot be earlier than the vehicle arrival time. The vehicle average waiting time at the container torage bloc i repreented in contraint (10). The precedence relationhip in the QC woring equence i guaranteed by contraint (11). Contraint (12) enure that the pic-up or drop-off ta of two paired container tart at the ame time. Contraint (13) define the calculation of maepan. Contraint (14) define the binary variable. 32

41 5. LOCAL SEQUENCE CUT ALGORITHM 5.1 Motivation The difficultie in olving the problem lie in the fact that when the number of container increae, the volume of binary deciion variable and the earch region expand quicly and ignificantly. The exact method, lie branch-and-bound method, cannot find the optimal, even a feaible olution within acceptable computation time. Thu, thi diertation propoe an algorithm to reduce the earch region by eliminating thoe feaible but undeirable olution. Then the olver earche for a good, or even the optimal olution, within a reduced feaible region. The propoed algorithm i inpired by the concept of logic cut. The algorithm reduce the feaible region by eliminating ome integer feaible olution demontrably not optimal, according to the logic conideration. Thee cut can be indeed very effective. They may ignificantly hrin the feaible region and coniderably quicen the reduction of the optimality gap throughout the iteration of the olver. A a reult, they can be extremely beneficial in reducing the CPU time in the earch for optimality. The concept of logic cut ha been uccefully applied in olving the complex combinational problem. Quadrifoglio et al. (70) developed et of logic cut to olve a MIP formulation for the tatic cheduling problem of a mobility allowance huttle tranit (MAST) ytem. They too advantage of thoe cut baed on the reaonable aumption of the paenger behavior to develop et of logic cut and impoe them into the MIP formulation. Indeed, they developed et of logic cut to peed up the earch for 33

42 optimality. The computation experiment revealed that the developed inequalitie effectively removed inefficient olution from the feaible region and reduced the computation time up to 90%. Guignard et al. (22) formulated an integrated timber harvet and tranportation planning problem a 0-1 MIP model. They ped the earching proce by uing different tightening technique, including the addition of logical inequalitie, lifting of inequalitie, and branch-and-bound branching prioritie baed on conideration of double-contraction. The algorithm propoed in thi diertation i named Local Sequence Cut (LSC) Algorithm. If container c i tranported immediately after g, c by the ame ', g' ' vehicle, then ( c, c ) i defined a a delivery equence of ', g ', g ' c g,. Thu, a olution to the problem i a combination of every container delivery equence. The entire feaible earch region i the et of all container feaible delivery equence atifying all contraint. Therefore, the miion of LSC algorithm i to reduce the earch region by eliminating thoe feaible but undeirable delivery equence without loing the poibility of finding a good, even optimal, integer olution. Since the objective of the problem i to minimize the maepan, it i reaonable that there hould be a time window for every QC ide ta tarting time. If a QC ide ta tart within it time window, it implie that the delivery of the container doe not delay the operation of that container or potpone the current bet maepan. Otherwie, it mean that the container i tranported too late or too early, and it current delivery equence i defined a a cut-off delivery equence and removed from the earch pace. 34

43 The upper and lower bound of thee time window are etimated through contructing and olving upper and lower ub-problem iteratively until all time window have been updated or the maximum number of iteration i reached. At the end of the algorithm, all the cut-off delivery equence baed on the final time window are excluded from the earch pace of the original MILP model by etting thoe correponding binary deciion variable to 0. Conequently, the olver earche for the olution within the reduced feaible region and i uppoed to find a good, even optimal olution much more quicly. The proce and detail of the algorithm are tated in the following. 5.2 Baic Scheme of Local Sequence Cut Algorithm At firt, we introduce the baic cheme of the propoed LSC algorithm. Denote the upper and lower bound of a QC ide ta tarting time, g a follow: y, Q, S, g 1 or 2 the earliet allowable tarting time of the QC ide, g y ta y ; or the lower bound of, g (,1,2 ) g, y y y, Q, S, g 1 or 2 the latet allowable tarting time of the QC ide ta, g y y ; or the upper bound of, g (,1,2 ) g, y y y 35

44 5.2.1 Initial Solution To tart the iteration, the algorithm require a et of initial lower and upper bound for the QC ide ta. The initial olution can be obtained by any method; we preent one imple approach here. We divide all vehicle into everal group a evenly a poible and each group erve one QC only. For intance, aume that there are three QC and ix vehicle, then all vehicle are divided into three group. Then the two vehicle in each group only erve one QC from the beginning to the end. FIGURE 9 how the initial aignment of vehicle and delivery equence of container. The container tranported equentially by Vehicle 1 are 1,1 2,1 3,1,1 S,1 {,,,,,, } c c c c c and thoe by Vehicle 2 are 1,2 2,2 3,2,2 S,2 {,,,,,, } c c c c c. A long the delivery equence are determined, each QC ide ta tarting time i nown at the ame time. Aume that the firt pair of container 1,1 c and 1,2 c unloaded by q are piced up by vehicle at time. Becaue of the precedence relationhip and the QC cycle time, the ucceeding container g, c operated by q cannot be piced up or dropped off by vehicle earlier than g, ( 1) (15) y The initial upper bound are calculated according to the objective value of the initial olution, viewed a the current bet maepan denoted by *. And the upper bound of all QC ide ta tarting time are calculated a: 36

45 g, * ( S ) (16) y The initial lower and upper bound calculated by equation (15) and (16) are quite looe and defined a the abolute lower and upper bound. Becaue there i no QC ide ta could tart earlier or later than it correponding abolute lower or upper bound. FIGURE 9 Initial delivery equence of the two vehicle in a group Time Window and Cut-off Delivery Sequence One core miion of the algorithm i to determine whether or not a feaible delivery equence of a container i a cut-off delivery equence, dependent on the relationhip between the QC ide ta tarting time and it time window. When container c g, i ', g ', g tranported following the equence ( c, c ), if the QC ide ta tarting time i ' later or earlier than the upper or lower bound of it current time window, then ( c, c ) i defined a a cut-off delivery equence. However, the tarting time are ', g ', g ' 37

46 unnown until the entire problem ha been olved. Thu, the algorithm ue an eay-tochec and comparative variable arrival time -- taing the place of tarting time in deciding cut-off delivery equence. Becaue the traveling time and the operation time at the YC ide are determinitic, the earliet and latet arrival time of a QC ide ta ( and, g ) are, eaily etimated by the following equation (17) and (18). After finihing it current QC y g y ide ta y ', g' ', the vehicle need ome time to arrive at it next QC ide ta y g,. It include the traveling time and neceary operation and waiting time at the YC ide. Here, we define the um of thoe time a etup time between the two ta ( ). ', g ', g y ', y Now the earliet and latet arrival time and, g, g can be calculated by adding the y y etup time to the lower and upper bound of ta equation (17) and (18). y g, tarting time a tated in η, g, ', g ' ', g ', g ' ' y y y,y η. g, ', g ' ', g ', g ' ' y y y,y (17) (18) ', g' The calculation of η ', g ', i determined by what type the ta y and y y,y ' g ' g, are. All the four poible cae are lited in TABLE 3. In cae 1, both ta y and ', g' ' y g, are pic-up ta at the QC ide, denoted by (P, P). The vehicle leave q ' after picing 38

47 up container c and then travel to the YC where container ', g' ' c i taced. After that, ', g' ' thi vehicle travel to the QC where container c g, i unloaded and pic it up. Thi whole proce, i.e., from QC to YC and then to QC, i denoted by Q-Y-Q in TABLE 3. One can eaily ee that η ', g ', y,y ' g conit of two traveling time and two handle time a hown in the equation (19). For the cae (P, D), the two adjacent QC ide ta are the pic-up and drop-off ta, repectively. The vehicle firt pic up the container c ', g' ' at q ' and drop it off at it detination bloc at the YC ide. Then thi vehicle pic up another container c at the YC ide and drop it off at q. Other cae in TABLE 3 can g, be explained in a imilar fahion. TABLE 3 Calculation of Setup Time Cae (P, P): Q Y Q (19) ', g ', g ', g ' ', g ' ', g ', g ', ', ' ', y y p d d p Cae (P, D): Q Y Y Q (20) ', g ', g ', g ' ', g ' ', g ', g, g, g ', ', ' ',, y y p d d p p d Cae (D, P): Q Q (21) ', g ', g ', g ', g ', ', y y d p Cae (D, D): Q Y Q (22) ', g ', g ', g ', g, g, g ', ',, y y d p p d Note: P: pic-up ta; D: drop-off ta; Q: quay crane; Y: yard crane. 39

48 Now we are ready to preent the criteria employed to determine the cut-off delivery equence. Illutrated in FIGURE 10 a an example, now the criterion depend on the relationhip between the earliet (latet) arrival time and the upper (lower) bound of the tarting time. Aume that time window of QC ide ta y m, gm m tarting time, where m 1 to 4, have been decided during earlier iteration of the propoed method and are [,, g, g ]. The range of thoe time window are ignified by the olid red bloc m m y m m m m y 2, g2 3, g3 4, g4 in FIGURE 10. To determine whether or not container c, c and c are allowed 1, 1 to be tranported immediately after container c g by the ame vehicle without violating their current time window, their earliet and latet poible arrival time [,, g] , g m m y m m m y m are calculated according to the equation (17) to (22) and preented by the white bloc. 1, g1 4, g4 According to the criteria, the delivery equence ( c, c ) i a cut-off delivery 1 4 equence, a the earliet arrival time i till later than the upper bound of the time 4, g4 window. The vehicle late arrival delay the operation of container c 4 and all it 2, g2 ucceeding operation and the current bet maepan. At the ame time, c hould not 1, 1 be tranported immediately after c g by the ame vehicle, either. It latet arrival time 1 2 i earlier than the time window lower bound, 2, g2 2 y 2 g y 2 2. Thi cae, of coure, doe not delay the operation of it ucceeding operation. However, the vehicle waiting time at the QC i a wate of vehicle reource and might affect other operation availabilitie of 1, g1 3, g3 vehicle, potponing the current bet maepan a a reult. Thu, only ( c, c ) i not

49 a cut-off delivery equence becaue it arrival time bloc overlap with the correponding tarting time bloc, i.e. 3, 3, 3,, 3, y g g 3 g 3 y 3, g 3 y 3 y In ummary, auming that a vehicle i dipatched to tranport the container c g, immediately after delivering c ', g' ', if the vehicle arrival time at the q violate the criteria expreed in (23), then the delivery equence ( c, c ) i a cut-off delivery ', g ', g ' equence.,,, g, g, g, g y y y y (23) FIGURE 10 Determination of cut-off delivery equence. 41

50 5.2.3 Sub-problem (a) Upper bound ub-problem (b) FIGURE 11 Contruction of ub-problem. Lower bound ub-problem 42

51 Now, the only iue left that critically entail the algorithm i the etimation of time window. Since the container delivery chedule i dependent on it delivery equence, a long a the earliet and latet delivery equence i determined, the lower and upper bound of the QC ide ta tarting time can be determined by contructing and olving ub-problem. In addition, the optimized reult of ub-problem are alo ued to chec whether or not the container earliet and latet delivery equence affect the operation of container and the current bet maepan. For the ae of illutration, we explain an upper bound ub-problem in detail. In FIGURE 11, thi ub-problem i to etimate 3,1 and y 3,2 2 y 2. Since thee two container 3 are operated in the ame tandem lift, there hould be 3,1 3,2. We define L a the et y2 y2 of the latet delivery equence of container c 3,1 and 3,2 2 c. According to the criteria in the 2 determination of cut-off delivery equence, if the container i tranported later than it latet delivery equence, the earliet arrival time will be later than the current time 3,1 window upper bound. A hown in FIGURE 11, if the container c i tranported 2 immediately after container c 4,1 by the ame vehicle, there will be 1 3,1 3,1 y y However, if 5,1 c i tranported immediately after container c then 3,1 3,1 (a violation). 3,1 2 1 y y 2 2 Conequently, the container 4,1 3 c i added to the et 1 L. Becaue the delivery chedule of 2 c i alo dependent on the delivery of c 3,2, operated in the ame tandem lift, the latet 3,1 2 2 delivery equence of 3,2 3 c are alo added into the et 2 L. 2 43

52 However, until now the latet delivery equence of container and are only defined according to the etimated time window and the calculated arrival time, intead of the actual tarting time. To chec whether or not the two target container latet delivery equence delay their operation or the current bet maepan, the algorithm optimize the delivery equence of all related container that are circled by the dahed frame, uing an upper bound ub-problem. In addition to the contraint of the original MILP model, there are ome extra contraint impoed on the MILP model of the ub-problem: (1) The two target container c 3,1 and 3,2 2 c2 mut be tranported following the latet 3 delivery equence in the et L by adding contraint 2, g 3 c L 2 2 x. g 1 g, 2 3, g d, p2 (2) The delivery equence of container repreented by the dahed grey bloc (denoted by 3 2 ) do not affect the delivery equence or chedule of related container, o the ub-problem doe not optimize thoe container delivery equence or chedule. Thu, they are tranported following their delivery equence of the initial olution. (3) To reduce the computation burden of the ub-problem, all temporary cut-off delivery equence of related container (denoted a the et 3 2 ) baed on current time window are eliminated from the earch pace by etting the correponding binary deciion variable to zero. They are defined a temporary cut-off equence becaue they are only valid in thi pecific ub- 44

53 problem. When contructing the next ub-problem, all the temporary cut-off equence need to be re-checed according to the new time window. After adding the extra contraint lited in TABLE 4, the algorithm contruct the upper ub-problem for 3,1 and 3,2 y 2 y 2 then olve it uing the branch-and-bound method. Aume that the optimized delivery equence of the target container are 1, g1 3,1 (, 2 ) 1 c c and 2, g2 3,2 (, 2 ) 2 c c and the tarting time of 3,1 y 2 and * y i 3,1. If * 3,2 3,1 3,1 3,1 ; thi 3,2 2 y 2 y y y y implie that their latet delivery equence do not violate the current upper bound of the time window or potpone the current bet maepan. Then the optimized tarting time * i ued a the new upper bound ( 3,1 31, ). In addition, all the container operated * y 3,1 2 y 2 y 2 before c 3,1 and 3,2 2 c2 update their upper bound at the ame time. Due to the precedence contraint and the QC cycle time, they could not be piced up or dropped off by any vehicle later than 3,1 (3 ) at the QC ide. y 2 If intead, * 3,2 3,1 3,1 3,1, it tate that even following the bet of the latet y y y y delivery equence, the deliverie of the two target container till delay the operation of container c 3,1 and 3,2 2 c2 o much that the current bet maepan i potponed. Then thoe 1, g1 3,1 two delivery equence ( c, c ) and 1 2 2, g2 3,2 (, 2 ) 2 c c are added to the et of permanent cut- off delivery equence (denoted a 2 ). Logically, ince it i too late to tranport the 3,1 3,2 container c or 2 c immediately after the container 1, g 1 2, g2 c and c 2 1 2, it i reaonable to forbid them or the container operated before them by q 2 to be tranported immediately 45

54 1, 1 after the container that are operated after c g 2, g2 or c 1 2. Furthermore, all thee correponding delivery equence are defined a the permanent cut-off delivery equence, a well. Ditinct from thoe temporary cut-off equence introduced above, the permanent cut-off equence are permanently removed from the earch pace of following ub-problem from thi point on. Note that thi i a equential heuritic approach and we are not guaranteeing that the overall optimal olution of the problem cannot include any one permanent cut-off delivery equence. Lower bound ub-problem are contructed and olved in a imilar fahion; indeed, the only difference prove to be whether or not the earliet delivery equence of target container affect other QC operation exceively, uch that the current bet maepan i potponed. Refer to FIGURE 11 (b): when container immediately after 3,1 c 2 i tranported 2,2 c 1 the latet arrival time i earlier than the lower bound of it current time window, i.e. 3,1 3,1. Meanwhile, if y y 2 2 3,1 c2 i tranported immediately after the ame vehicle, the vehicle arrival time doe not violate the current lower bound of the tarting time, i.e. 3,1 3,1. Then the container y y 2 2 3,2 c1 3,2 c1 by i added into the et of the earliet delivery equence F. After determining the earliet delivery equence of 3 2 3,1 c 2 and 3,2 c 2, the lower bound ub-problem i contructed through addition of extra contraint (28)-(31) to the original MILP model. If the optimized reult of the lower bound ub-problem reflect that the target container earliet delivery equence do not delay other QC operation, the optimized 46

55 tarting time i employed to update the lower bound of the target container QC ide ta tarting time. At the ame time, the container operated after thee target container update their lower bound of tarting time. Otherwie, if the reult tate that the target container are tranported too early and that other QC operation are delayed o much that the current bet maepan i potponed a a reult, their earliet delivery equence are defined a permanent cut-off delivery equence. The container operated after thee target container cannot be tranported earlier than thoe earliet delivery equence. TABLE 4 Extra Contraint Impoed to a Sub-Problem Extra contraint added to the upper bound ub-problem of, 2, g c L g 1 x x, g ', g ', p ' d, g ', g ', p ' d x g, 2, g d, p c c, g ', g ' ' g y (24) 0,, c (25) 1,, c (26), g ', g ' ', Q, S, g y Extra contraint added to the lower bound ub-problem of, 2, g c F g 1 x x, g ', g ', p ' d, g ', g ', p ' d x, g, g d, p 2, g ', g ' ' g y (27) (28) 0, c, c (29) 1, c, c (30), g ', g ' ', Q, S, g y (31) 47

56 5.2.4 Analyi of the Sub-problem Reult Due to time contraint, not every ub-problem can be olved to optimality. Therefore, it i important to analyze the reult and extract a much information a poible to update the time window and the permanent cut-off delivery equence. Thi procedure i referred to a the analyi of the ub-problem reult. The firt goal of the proce i to determine whether or not the olution i eligible for further conideration. If,1,2 it i, the analyi i to determine whether or not the target container c and c are allowed to be delivered following the current optimized delivery equence without worening the current optimal maepan. The time window or the et of permanent cutoff delivery equence are updated accordingly; otherwie, the reult are dicarded directly. The complete proce i preented in TABLE 5 and TABLE 6 Upper Bound Sub-problem The core miion of the upper bound ub-problem i to chec whether or not the target,1 container c and c,2 potpone current optimal maepan. latet delivery equence are late, delay the operation of q, and Aume that the optimal olution to the ub-problem i, *. The reult 1, g1,1 obtained within the time limit i g, and the current delivery equence are ( c, c ) y 2, g2,2 and ( c, c ). If the ub-problem ha been olved to optimal within the time limit, i.e. 2 g y 1, g, g * y y, the algorithm update the time window or the et of permanent cur-off 48

57 delivery equence following the rule introduced in the lat part. If the ub-problem i not olved to the optimal within the time limit, it i poible that, g, g *. Therefore,,1,2 even if, g, g hold, we cannot conclude that the container c and c are y y tranported too late following their latet delivery equence. To avoid the excluion of potential good olution due to the poibly inaccurate etimation of actual tarting time and time window, an alternative criterion i utilized in maing the deciion. Note that y y * ( S ) i the abolute upper bound for any QC ide ta y g, becaue of the precedence relationhip. For the olution to the upper bound ub-problem, there are three cae: (1), * ( S ) g y g, ha exceeded it upper bound o much that the maepan i necearily delayed. y The current upper bound, hall not be updated baed on thi optimization reult. g y 1, g1,1 2, g2,2 However, whether or not the current delivery equence ( c, c ) and ( c, c ) hould be cut forever from the earch pace a permanent cut-off equence till require more dicuion. 1, 1 (a) If the tarting time of y g 2, g2 or y i alo later than it abolute upper bound ( * ( S ) or, g * ( S ) ), it i highly poible that the ize 1, g y y of the ub-problem i too large to be olved to optimal within the hort time limit. Hence, the reult are viewed a ineligible for any update. 49

58 1, 1 (b) If both y g 2, g2 and y tart earlier than their abolute upper bound ( 1 2 * ( S ) and, g * ( S ) ), the upper bound ub-problem 1, g y y i viewed a olved to near-optimal. The olution may be very cloe, or even the ame a the optimal olution. In uch a cae, we determine that it i too late to,1,2 tranport the container c and c in the current delivery equence and their ucceeding operation cannot be finihed before their abolute upper bound. Then the algorithm update the et of permanent cur-off delivery equence accordingly. (2), * ( S ) g y g, doe not exceed it abolute upper bound and the maepan i not necearily y delayed. Thu the current delivery equence are not defined a the permanent cutoff equence. However, whether or not the optimized tarting time can be ued to update, till neceitate further dicuion. g y 1, 1 (a) If both y g 2, g2 and y tart later than their abolute upper bound, it i highly 1 2 poible that the ize of the ub-problem i too large to be olved to optimal within the time limit. Hence, the reult are viewed a ineligible to update the time window. 1, 1 (b) If at mot only one ta, y g 2, g2 or y, tart later than it abolute upper bound, 1 2,1,2 it implie that the current bet delivery equence of the container c and c do 50

59 not necearily potpone the current bet maepan * ; thu, the algorithm update the upper bound uing the optimized tarting time, i.e., g,. At the ame time, all the container operated before the target container by q hould chec, and update if neceary, the upper bound of their QC ide ta tarting time according to the precedence relationhip uing equation (32). y g y min(, ), ' (32) ', g ', g ' 1, g y y y (c) If * and all container delivery equence have been optimized in the ubproblem, i the maepan of the entire problem. Thu, if the maepan obtained from the optimization reult i better than the current bet olution *, then *. TABLE 5 Analyi of an Upper Bound Sub-problem Analyi of the upper bound up-problem of, if, * ( S ) then g y 51 g y if, g * ( S ) and, g * ( S ) then y go to the next ub-problem; ele, g, g y y 1 y

60 TABLE 5 Continued min(, ), ' ', g ', g ' 1, g y y y if * and all container delivery equence have been optimized in the ubproblem then * ; min(, ) where ' Q, ', ' 1 S ' ; end if end if end if ', g ' ', g ' ' 1, g ' y ' y ' y ' if, * ( S ) then g y if, g * ( S ) and, g * ( S ) then y remove 1 1 ', g ', g (, ) 1 y c c from the earch pace a permanent cut-off delivery equence, where ' and ' ; 1 remove ', g ', g (, ) 2 c c from the earch pace a permanent cut-off delivery equence, where ' and ' ; 2 ele end if go to the next ub-problem; end if Lower Bound Sub-problem For a lower bound ub-problem, the optimization reult i to decide whether or not the,1,2 container c and c are tranported too early uch that the other QC operation are delayed due to lac of acceibility to the vehicle. Now, aume that the lower bound 52

61 ub-problem of, raied are: (1), g, y g y g y ha been olved within a hort time limit. The poible ituation According to the principle in contructing the lower bound ub-problem, thi cenario hould be very rare. It i highly poible that the ize of the ub-problem i too large to be olved to optimal in the time limit. Therefore, the reult are ignored without any further analyi. Here we ue g, intead of * ( S ) to avoid the ituation that might reult in the mitae in eeing cut-off delivery equence. (2), g, y g y 1, 1 (a) If both y g 2, g2 and y tart later than their abolute upper bound ( 1 2 y * ( S ) and, g * ( S ) ), the implication i that 1, g y y , container c are tranported too early and that other QC operation are delayed g o much that the current optimal maepan i potponed a a reult. Thu, the algorithm update the et of permanent cut-off delivery equence according to the reult. (b) the implication i that the current delivery equence do not affect the operation of q 1 or q. Hence, the reult are eligible to update the lower bound, i.e., 2 g, g, y y. At the ame time, all the container operated after the target container chec and update their lower bound according to the equation (33) 53

62 max(, ) where ' (33) ', g ' 1, g ', g y y y TABLE 6 Analyi of a Lower Bound Sub-problem Analyi of the lower bound ub-problem of, g y if, g, g y y then ele if 1, g 1 1, g y y end if or,, ; g, y g, y 2 g2 2 g y y ', g ' 1, g ', g then max(, ) where ' end if y y y ; if, g * ( S ) and, g * ( S ) then y remove 1 1 ', g ',1 (, ) 1 y c c from the earch pace a permanent cut-off delivery equence, where ' and ' ; 1 remove ', g ',2 (, ) 2 c c from the earch pace a permanent cut-off delivery equence, where ' and ' ; 2 end if go to the next ub-problem; 54

63 5.2.5 Proce of Baic Scheme Combining the part introduced above, the baic cheme of the propoed LSC algorithm i illutrated in TABLE 7. TABLE 7 Proce of the Baic Scheme for Local Sequence Cut Algorithm The Baic Scheme for Local Sequence Cut Algorithm for each tranport requet r do ee the temporary cut-off delivery equence that violate the current time bound and determine the et L,, and ; contruct the upper bound ub-problem of g, ; olve the ub-problem; analyze the reult of the upper bound ub-problem; end for for each tranport requet r do ee the temporary cut-off delivery equence that violate the current time bound and determine the et F,, and ; contruct the lower bound ub-problem of g, ; olve the ub-problem; analyze the reult of the lower bound ub-problem; end for introduce all cut-off equence into the original MILP model and et all correponding deciion variable x to be 0; olve the original MILP model. y y 55

64 5.3 Enhanced Scheme A the problem ize increae, even olving the ub-problem become time-conuming. In addition to the baic cheme, it i neceary to develop everal enhanced cheme for improving the performance of the algorithm Treatment at Early Stage During the firt iteration, the time window are uually too looe to generate enough cut-off delivery equence, and there are alway too many container that can be delivered before or after the target container without violating the time window. A a reult, the ize of the ub-problem i alway too large for the exact method. Thu, it i reaonable to regulate that the delivery equence g, g, (, ) c c to be a cut-off equence if ab ( ) floor( V / Q ) (34) Size Limit on Sub-problem If the ize of a ub-problem i too large for the MILP olver, the olver may not find any feaible integer olution or only find a bad feaible olution whoe reult i not eligible for any update. To avoid either of the cae, the ize of the ub-problem hould be examined before it contruction. For each ub-problem in the baic cheme, a threhold i impoed on the number of the deciion variable that have been decided according to 56

65 the et and to the next ub-problem.. If the minimum threhold cannot be met, the algorithm directly ip One hould note that the upper and lower bound of tarting time have different effect on earching olution. A relatively higher upper bound doe not exclude a deirable delivery equence a a cut-off equence while a relatively higher lower bound may remove good equence in the ucceeding earching procee. To guarantee a higher quality of the olution, a tricter limit i impoed on the problem ize for the lower bound ub-problem than that for the upper bound ub-problem. Moreover, we point out here that becaue the computation peed of the LSC algorithm i greatly determined by the update of time window, the computation might be accelerated along with the increaing vehicle. 5.4 Overall Proce of LSC Algorithm After adding the enhanced cheme, the overall proce of LSC algorithm i tated in TABLE 8. TABLE 8 Entire Proce of Local Sequence Cut Algorithm input initial upper and lower bound; do while not all the g,, Q, S have been updated or the maximum number of iteration ha not been reached; for each tranport requet y r do 57

66 TABLE 8 Continued determine the et L, and ; if the ub-problem ize i maller than the threhold then contruct the upper bound ub-problem of g, ; olve ub-problem within a time limit; if the olver cannot find any feaible olution then ele go to the next ub-problem; analyze the reult of the upper bound ub-problem; end if end if end for for each tranport requet determine the et r do F, and ; if the ub-problem ize i maller than the threhold then contruct the lower bound ub-problem of g, ; olve the ub-problem within a time limit; if the olver cannot find any feaible olution then go to the next ub-problem; ele analyze the reult of the lower bound ub-problem; end if end if end for end do while introduce all cut-off equence into the original MILP model and et all correponding deciion variable x to be 0; olve the MILP model within a time limit; if * then 58 y y

67 TABLE 8 Continued * end if 5.5 Numerical Experiment In thi ection, we evaluate the performance of LSC algorithm againt the branch-andbound method realized by CPLEX. All the run are performed uing CPLEX 12.1 with default etting and C++ in a 3.30 GHz CPU with 4.0 GB RAM. The QC cycle time ue the mode value in the empirical ditribution lited in TABLE 1. The handle time at the QC/YC include the neceary time that a QC/YC lift/retrieve a container onto/from a vehicle, a well a the extra travel time becaue of the vehicle deceleration a it approache and leave a QC/YC (39). The average waiting time of vehicle at the YC ide ue the hitorical value in the NCT, alo conitent with the value in the tudy of Lee and Kim (46, 47). It include the waiting time of the pa permit before the vehicle enter the paenger lane and the poible waiting for availability of the YC LSC v.. Optimality To illutrate the LSC method and tet it performance, thi ection preent a erie of numerical experiment. To allow for tatitically meaningful reult, for each problem ize 10 cae were executed. The reult are preented herein. 59

68 We compare the performance of the LSC method againt the MILP model, directly olving by the ILOG-CPLEX in term of objective value and the computation time. To enure that the CPLEX find the optimal olution, we firt compared the two method in twenty mall ize intance, lited in TABLE 9. The cale of a cae i repreented by the number of the cae number. For example, implie that there are in total twenty-four container, 48 pic-up and drop-off ta, and nine vehicle in the model; and the lat number 1 denote the firt cae of the ame problem ize. All the woring equence of QC and the torage bloc of container are randomly generated by the program. The econd and third column preent the objective value found by the CPLEX and the propoed LSC method, repectively. The reult lited in TABLE 9 indicate that the LSC method i able to find the optimal olution in mot cae. For the cae with twenty-four container and 9 vehicle, both method olve the MILP model very quicly. The problem i olved to optimal via the LSC method in 9 out of 10 cae. Even in the only exception, i.e., cae 1, the olution gap i marginal compared to the optimal one. When there are 36 container and 12 vehicle in the model, the number of binary deciion variable increae ignificantly from 595 to Thu, the CPLEX need up to 1.5 hour to olve the problem to optimal. However, the propoed LSC method olve the problem within 15 minute and find the optimal olution within uch a hort time in 6 out 10 cae. The larget gap between the optimal olution i only about 3.4% and the average gap i maller than 1%. 60

69 Next, the propoed LSC algorithm i computationally teted in a large tet bed, where the intance are conidered rigorouly, uing exact method. In TABLE 10, the computational time of the CPLEX and LSC method are illutrated in the column with the name T_CPLEX and T_LSC, repectively. We allow the CPLEX to wor for a time limit up to 10 hour. The bet integer olution found by the CPLEX within the time limit are ued a the benchmar for aeing the LSC algorithm. The objective of the olution and computational time of the LSC method are revealed in the column named LSC and T_LSC, repectively. TABLE 9 LSC v.. Optimality (Small Size Problem) Cae CPLEX LSC Gap (%) average

70 TABLE 9 Continued Cae CPLEX LSC Gap (%) average Note: CPLEX the optimal olution found by CPLEX-OLOG; LSC the bet olution found by LSC method; Gap (LSC CPLEX) / CPLEX * 100% In 16 out of 40 cae, the LSC method find a better olution than the CPLEX. The objective value obtained from the LSC method i % lower in the bet cae and 6.397% higher in the wort cae, compared with the CPLEX. The average gap between the two method doe not exceed 4%. However, the computation time conumed by the LSC method i only 1/5 to 1/4 of the CPLEX. TABLE 10 LSC v.. Optimality (Large Size Problem) Cae No. Bet Integer Bet Node T_CPLEX LSC T_LSC Gap hr min hr min hr min hr min hr min hr min hr min hr min hr min hr min average hr min hr min

71 TABLE 10 Continued Cae No. Bet Integer Bet Node T_CPLEX LSC T_LSC Gap hr min hr min hr min hr min hr min hr min hr min hr min average hr min hr min hr min hr min hr min hr min hr min hr min hr min hr min average hr hr hr hr hr hr hr hr hr hr hr hr hr hr hr hr hr hr hr hr average Note: Bet Node the bet non-integer olution found by the CPELX 63

72 Another important obervation i that for the problem that conit of the ame number of container, the more vehicle there are, the fater the problem i olved. The main reaon i becaue more vehicle lead to the relatively lower upper bound of tarting time. Since the contruction of a ub-problem i dependent on the range of time window, the lower upper bound i undoubtedly helpful in etimating the tarting time more preciely and controlling the ize of ub-problem effectively. Conequently, they accelerate the olution of ub-problem and horten the total CPU time of the algorithm Time Window and Cut-off Delivery Sequence At the end of the algorithm, the CPLEX i allowed to olve the original MILP model within only 20 minute. It olution i comparable to the one found by the CPLEX alone running 10 hour. The ucce i attributed to the introduction of thoe cut-off delivery equence defined by the algorithm. According to the numerical experiment, at the end of the algorithm, the binary deciion variable of the original MILP model have been halved before the CPLEX tart to olve it. To tate how the cut-off delivery equence and the time bound are updated through iteration, we tae one cae a an intance. FIGURE 12 plot the number of cut-off delivery equence eliminated from the earch pace of the ub-problem. They are compried of thoe temporary cut-off delivery equence according to the current time window and the permanent cut-off delivery equence decided during earlier iteration. Here, we only record the ubproblem whoe ize atify the pre-defined threhold, a only thee would proceed to 64

73 be olved by the CPLEX. There are a total of 151 ub-problem contructed and olved during iteration. We notice that the number of cut-off delivery equence alway eep increaing when the algorithm i olving upper bound ub-problem. When the algorithm tart to contruct and olve lower bound ub-problem, the number of cut-off delivery equence reache relative tability or increae very lowly. Thi implie that the update of tarting time lower bound i leer than upper bound. That i becaue (1), in contructing a lower bound ub-problem, the target container are forced to be tranported following their earliet delivery equence. A a reult, the optimized tarting time of the correponding QC ide ta i the ame a it current lower bound; and (2), to avoid excluding potential good delivery equence from the earch pace, the criteria in updating time window or the et of permanent delivery equence i tricter than analyi of the reult of upper bound ub-problem. FIGURE 12 implie that the upper bound ub-problem are more critical to the update of time window and the et of permanent cut-off delivery equence. To preent how the time window are updated through iteration, FIGURE 13 record the time window of all container operated by the ame QC. Becaue the two container operated in the ame tandem lift have the ame time window for their QC ide ta, each bar in FIGURE 13 repreent the time window of thoe two QC ide ta tarting time. The line above and below the bar are the upper and lower bound of the time window. 65

74 # cut-off ub-problem FIGURE 12 Number of cut-off delivery equence through iteration. FIGURE 13(a) i the original time window calculated uing the initial olution and the QC cycle time. The initial time window are very looe and they hare the ame range. During the firt iteration, only the firt few time window are updated, while other remain the ame, becaue mot ub-problem are not olved by the CPLEX due to their problem ize. When more time window are updated, more ub-problem atify the pre-defined threhold and are olved by the CPLEX. FIGURE 13(c) (d) how very clearly that the time window are narrowed greatly through the next two iteration. When the third iteration end, all tarting time have been updated. The green triangle plotted in FIGURE 13(d) repreent the tarting time in the final olution, all viibly poitioned within the final time window. It prove that the effectivene of LSC algorithm in etimating the time window. 66

75 Time Bound Time Bound Time Bound Operation (a) The initial time window Ta (b) After the firt iteration Ta (c) After the econd iteration FIGURE 13 Update of time bound through iteration. 67

76 Start Time FIGURE 13 Continued Ta (d) After the third iteration 68

77 6. MODIFIED LSC ALGORITHM The numerical experiment in the lat ection have hown that the propoed LSC algorithm i capable of reducing the earch pace effectively by cutting off the delivery equence baed on the etimation of time window. However, the original algorithm i till not efficient enough to handle the problem of larger ize. The reaon can be ummarized a follow: (1) According to the analyi in previou ection, the update of upper bound play a ey role in the algorithm. It determine the quality and the peed of the algorithm ignificantly. Unfortunately, the upper bound of many tarting time cannot be updated at the beginning tage. Such a diadvantage i more crucial when the operation increae. The algorithm need more iteration to update the time window of all QC ide ta tarting time. (2) Contrary to thoe ub-problem not contructed or olved due to their large ize, there are ome other ub-problem are contructed and olved repeatedly through iteration without contributing any new information to the algorithm. The et of the latet or earliet delivery equence are the ame a thoe in the lat iteration; the optimized tarting time of ub-problem i the ame a the current upper or lower bound. In uch cae, thee ub-problem are defined a meaningle ub-problem, becaue the algorithm doe not benefit from their olution but conume CPU time only. 69

78 To olve the above two problem and further accelerate computation, thi ection i dedicated to propoe the modified LSC algorithm. The olution to the firt problem i to propoe a heuritic method a an alternative to the CPLEX to olve the ub-problem, epecially when the ub-problem ize doe not atify the pre-defined threhold. In addition, it computation time hould be more competitive than the CPLEX and not affected ignificantly by the problem ize a the CPLEX. The olution to the econd problem i to develop a et of election mechanim to prevent the algorithm from contructing and olving thoe meaningle ub-problem. In ummary, the employment of the heuritic method i to accelerate the update of the time window, epecially the upper bound, of the tarting time. The introduction of the election mechanim i to detect the meaningle ub-problem and prevent wating CPU time on them. Thu, the modified LSC method i expected to quicen the earching proce ignificantly compared to the original algorithm, without affecting the olution quality. The following part introduce the modified LSC method in more detail. 6.1 Priority Baed Heuritic Method To provide a more computationally efficient method to olve the ub-problem, a heuritic method i propoed and ued to olve the upper bound ub-problem without any ize limit. If the reult i eligible to update the upper bound of the tarting time, it will not be olved by the CPLEX again even if it ize atifie the pre-defined threhold. To avoid the elimination of good olution, the reult from the heuritic method would not be ued to define the permanent cut-off equence, even if it i 70

79 eligible for that. Baed on the imilar conideration, the heuritic method i not ued to olve the lower bound ub-problem. In addition, to accelerate the computation further, a long a one time window i updated, the algorithm olve the entire problem via the heuritic method. If the objective i better than the current bet maepan, it i ued a the current bet maepan, and every tarting time upper bound i checed and updated if neceary according to the equation (16). The heuritic method only conider the active tranportation requet in maing the deciion. An active requet refer to the one whoe predeceor requet have all been completed, and we can predict when the container i ready to be piced up or dropped off at the QC ide. For example, if q ( 1) th operation i a loading (unloading) operation, then the requet r i releaed and become active when q finihe lifting (poitioning) the container from (onto) the vehicle. Thu, at any moment, there are at mot Q active requet, and each of them need two vehicle to tranport the container. An aignment of a tranport requet r to two vehicle vm and v i denoted a a( r, v, v ) here and after. n m n The miion of the algorithm i to determine which one i the bet among all poible aignment. To deduce the aignment that benefit the mot to minimize the maepan, a priority-baed heuritic method i developed to meaure the prioritie of the aignment by comparing their attribute. Then the vehicle are dipatched according to the aignment with the highet priority. The conideration of the priority meaurement 71

80 include the minimization of the QC idle time and maximization of the vehicle utilization Creation of Aignment An active requet r need two vehicle to tranport the target container,1 c and,2 c. The principle in contructing ub-problem are till applied in creating the aignment. 1, g1 2, g2 If c, c i a temporary cut-off delivery equence according to the current time 1 2 window or a permanent cut-off delivery equence according to the previou iteration, 1, 1 then the vehicle that i delivering container c g 1 cannot be dipatched to tranport 2, g2 container c conecutively. At the ame time, the two target container 2,1 c and mut be tranported following one of their latet or earliet delivery equence. All atifying aignment of the requet arrival time at a QC. In uch a cae, both the QC and the vehicle, which 72,2 c r are added to the et. For the purpoe of priority meaurement, the following attribute of each aignment are calculated and recorded: (1) QC idle time denote the QC waiting time due to the late arrival of vehicle. When the vehicle arrive later than the time when a QC i ready to lift/poition container from/onto vehicle, the QC ha to wait for them. It i obviouly that the minimization of a QC idle time i helpful to horten the maepan and increae the productivity of the QC. (2) Arrival time difference denote the difference between the two vehicle

81 arrive earlier, have to wait for the other vehicle. It not only reult in the QC idle time but alo affect the utilization of vehicle. (3) Empty travel time denote the travel time of the vehicle when it i empty. After dropping off a container at the YC or QC ide, the time an empty vehicle tae to travel to pic up the next container i defined a it empty travel time. (4) Arrival time difference denote the difference between the two vehicle arrival time at a QC. In uch a cae, both the QC and the vehicle, which arrive earlier, have to wait for the other vehicle. It not only reult in the QC idle time but alo affect the utilization of vehicle. (5) Free time denote the moment when the vehicle finihe dropping off it current container at the QC or YC ide and become free to repond another tranportation requet. (6) Remaining requet denote the number of ucceeding requet after according to q woring equence. The more remaining requet an active requet ha, the more it delay would affect that QC ucceeding operation and the final maepa. r Priority Meaurement After the contruction of aignment, the priority meaurement i conducted in two tep. In the firt tep, the comparion i among the aignment with the ame active 73

82 requet and only the bet one are ept for the further conideration. The QC idle time and arrival time difference are choen for the priority meaurement in the firt tep. They are the two attribute that affect the maepan directly and reflect the characteritic of the tandem lift QC. The aignment with the hortet QC idle time and the mallet arrival time difference are favored over other. The priority meaurement in the firt tep i illutrated in TABLE 11. TABLE 11 Priority Meaurement in the Firt Step if 1 eep the aignment with the hortet QC idle time then delete other; end if if 1 eep the aignment with the hortet arrival time difference then delete other; end if After the comparion and filtration in the firt tep, only thoe bet aignment are preerved for the meaurement in the econd tep. However, the ame vehicle may appear in the bet aignment of different requet. To mae the final deciion, all aignment of different active requet are compared and et prioritie in the econd tep. The choice of attribute and the order in which the attribute are compared mainly focu on the maximization of the vehicle utilization and the degree to which the attribute affect the maepan. If more than one aignment ha the equally highet 74

83 priority at the end of the econd tep, the algorithm pic one randomly from them. The priority meaurement in the econd tep i preented in TABLE 12. TABLE 12 Priority Meaurement in the Second Step if 1 then Q eep the aignment that have the hortet empty travel time and delete other; end if if 1then Q eep the aignment that have the mot remaining requet and delete other; end if if 1then Q eep the aignment that have the hortet QC idle time and delete other; end if if 1then Q eep the aignment that have the earliet free time and delete other; end if if 1then Q eep the aignment that have the mallet arrival time difference and delete other; end if if 1then Q eep the aignment that have the mallet early arrival time and delete other; end if if 1then Q pic up one aignment randomly; end if if 1then Q dipatch vehicle according to the aignment with the highet priority; end if 75

84 Aume that the aignment, m, n a r v v ha the highet priority and the vehicle have been dipatched accordingly; the next requet 1 r i releaed and become active. Then the heuritic method repeat the proce until the two target container, whoe tarting time upper bound i the concern of the ub-problem, have been tranported. If the reult i eligible for the update of the upper bound, the ubproblem will not be olved by the CPLEX again. Otherwie, the ub-problem i ent to the CPLEX if it ize atifie the pre-defined threhold. 6.2 Selection Mechanim To avoid the wate of CPU time on the meaningle ub-problem, a election mechanim i propoed in the modified algorithm. The mechanim i triggered before the contruction of a ub-problem. In the th n iteration, the upper bound ub-problem or lower bound ub-problem will be viewed a a meaningle ub-problem when all the following criteria are met: (1) In the ( n 1) th iteration, the upper (lower) bound ub-problem ha been olved by the CPLEX and it reult wa eligible for the update of upper (lower) bound of the time window. (2) In the ( n 1) th iteration, the optimized tarting time wa the ame a the upper (lower) bound before the contruction of the ub-problem. (3) In the th n iteration, the et of the latet (earliet) delivery equence of the target container are the ame a thoe in the ( n 76 1) th iteration.

85 If an upper bound or lower bound ub-problem i detected to be a meaningle ubproblem, the algorithm ip to the next ub-problem directly without contructing or olving it at all. The overall proce of the modified algorithm i illutrated in TABLE 13. TABLE 13 Proce of Modified LSC Algorithm for each tranport requet r do if the upper bound ub-problem of g, i not a meaningle ub-problem then determine the et L, and ; contruct the upper bound ub-problem; olve the ub-problem uing priority-baed heuritic method; analyze the reult of the upper bound ub-problem; y if the reult i not eligible to update the upper bound and the ub-problem atifie the pre-defined threhold then olve the ub-problem uing the CPLEX; analyze the reult of the upper bound ub-problem; end if end if end for olve the entire problem uing priority-baed heuritic method baed on the current upper and lower bound; if * then * end if for each tranport requet r do if the upper bound ub-problem of g, i not a meaningle ub-problem then y 77

86 TABLE 13 Continued determine the et F, and ; contruct the lower bound ub-problem; olve the ub-problem; analyze the reult of the lower bound ub-problem; end if end for olve the problem uing priority-baed heuritic method baed on the current upper and lower bound; if * then * end if add all cut-off equence into the original MILP model and et all correponding deciion variable x to be 0; olve the original MILP model uing the CPLEX 6.3 Numerical Experiment Reult To tet the effectivene of the propoed heuritic method and the election mechanim, a erie of numerical experiment are conducted. The olution and the computation time of the modified algorithm are compared with the original algorithm introduced in the lat ection and the branch-and-bound method realized uing the CPLEX. 78

87 TABLE 14 Numerical Experiment Reult of Different Method CPLEX LSC Mod LSC Gap (%) Obj Obj Obj time Cut (%) CPLEX LSC min min min min min min min min min min average min min min min min min min min min min average min min min min min min min min min min average min min min

88 TABLE 14 Continued CPLEX LSC Mod LSC Gap (%) Obj Obj Obj time Cut (%) CPLEX LSC min min min min min min min average Note: Cut (%) percentage of deciion variable that have been et to be 0 according to the cut-off delivery equence defined by the algorithm Gap/CPLEX (%) (Mod LSC CPLEX)/CPLEX*100% Gap/LSC (%) (Mod LSC LSC)/LSC*100% The reult lited in TABLE 14 how clearly that the propoed heuritic method and the election mechanim are effective in aving CPU time greatly, a expected. The computation time conumed by the modified LSC algorithm i only 1/4 to 1/3 of the original algorithm. More important i that when the deciion variable increae greatly, from around 5250 to around 8200, the computation time of the modified algorithm doe not increae a ignificantly a the original. It prove that the modified algorithm ha more potential and i more appealing in handling larger ize problem. To ae the effectivene of the algorithm in tightening the earch pace, we recorded the number of binary deciion variable that have been et to be 0 according to the cut-off delivery equence at the end of the algorithm. The reult lited in TABLE 14 indicate that the earch pace of the original MILP model i halved. Thu, the CPLEX find the equally 80

89 good or even better olution in 20 minute compared with the olution found by the CPLEX alone running 10 hour. Compared to the bet integer olution found by the CPLEX alone, the objective value obtained from the modified LSC method i 3.649% higher in the wort cae and % lower in the bet cae. Compared with original one, the modified algorithm i 8.333% higher in the wort cae and 6.284% lower in the bet cae. On average, the gap between the modified LSC algorithm, the branch-and-cut method, and the original algorithm i within 3% and 1%, repectively. In approximately half of thoe forty cae, the modified LSC method render a better olution than other two method in a much horter CPU time. When the problem ize eep extending, the CPLEX alone could not find a feaible olution, even running up to 10 hour. Thu, the abolute lower bound (LB) i choen for examining the performance of the modified LSC algorithm in the next 10 cae. The LB i calculated auming that there i no idling and delay during the QC operation. Thu, the optimal olution i impoibly lower than the LB provided here. The total computation time conumed by the modified LSC method i 30 minute, and 10 minute are conumed to olve the ub-problem iteratively. According to the reult lited in TABLE 15, the number of deciion variable i reduced by about 45% becaue of the introduction of cut-off delivery equence. With a much lower computation load, the CPLEX find a good olution in only 20 minute. And it average gap between the LB doe not exceed 15.4%. 81

90 TABLE 15 Modified LSC Algorithm v.. LB Mod. LSC time # bin Cut (%) LB Gap (%) min min min min min min min min min min average Note: # bin number of binary deciion variable Cut (%) percentage of deciion variable that have been et to be 0 according to the cut-off delivery equence defined by the algorithm 6.4 Senitivity Analyi To ae the robutne of the algorithm, enitivity analyi i conducted to ee whether or not the earch proce i greatly affected by the important parameter: initial olution and the time limit. The initial olution determine the initial upper and lower bound of all tarting time and therefore determine the contruction of ub-problem during iteration. The time limit in olving ub-problem might affect the optimization of ub-problem and thu the update of time window and the et of permanent cut-off delivery equence. The number of container and vehicle in the following experiment are et to be 108 and 21, repectively. 82

91 6.4.1 Initial Solution A decribed, we chooe a very imple method to generate the initial olution to tart the iteration. However, if the algorithm i enitive to the initial olution, we have to reconider the method of obtaining the initial olution or even the deign of the algorithm. To tet the enitivity of the algorithm to the initial olution, we increae and decreae the initial olution by 10% and 20% to determine whether the final olution would be ignificantly affected by the change. Each objective value lited in TABLE 16 i the average of 10 cae. TABLE 16 Algorithm Senitivity to the Initial Solution Initial Solution -20% -10% 0% 10% 20% -20% -10% Objective The computation reult preented in TABLE 16 how that the change in the initial olution do not affect the objective value. When the objective value of the initial olution decreae and increae by 10% and 20%, the objective value only change around 1% and 2.5%. Additionally, the total CPU time i not affected. Thi implie that the algorithm i robut and not enitive to the initial olution. The iteration can tart with an initial olution obtained from any imple and traightforward method. 83

92 6.4.2 Time Limit for Solving Sub-problem The longer the CPLEX i allowed to olve a ub-problem, the more poible that the ub-problem i olved to optimal, and the more accurate the upper and lower bound could be. However, the price for that accuracy i the poible increae in the CPU time, epecially when the problem ize expand. To analyze to what extent the time limit affect the algorithm, we re-calculate thoe 10 cae, varying the time limit in olving ub-problem from 40 to 100 econd. TABLE 17 Algorithm Senitivity to the Time Limit in Solving Sub-problem Time Limit (ec) Objective The reult lited in TABLE 17 reveal that when the time limit increae from 40 to 100 econd, there i only a minor decreae in the objective value. The gap between the lowet and the highet average of objective value i only 2.12%. Additionally, the computation time alo remain the ame when the time limit varie from the 50 to 90 econd. The reaon that the algorithm i not enitive to the time limit in olving ubproblem can be ummarized a follow: (1), becaue of the introduction of cut-off delivery equence, mot ub-problem can be olved to optimal within a hort time limit. Thu, the change of the time limit doe not affect the bet olution and the 84

93 computation time of mot ub-problem, and (2), even if the ub-problem i not olved to optimal due to the reduction of the time limit, the analyi of ub-problem reult prevent thoe bad reult from being ued to update the time window or the et of permanent cut-off delivery equence. In um, the enitivity analyi prove that the algorithm i robut. It computation time and objective value are not affected by the initial olution or the time limit in olving ub-problem. In addition, the comparion between the original and modified LSC algorithm ha hown that the application of heuritic method a the ubproblem olver quicen the earching proce greatly. Thu, it i reaonable to believe that the propoed algorithm would be improved further if another more effective heuritic method i employed to olve the ub-problem and the original MILP model. 85

94 7. ON-LINE DISPATCHING POLICY In addition to cloely interrelated activitie, the high degree of complexity of the terminal operation i due to the terminal dynamic woring environment. They include weather condition; wrong information of container and external vehicle; the mitae during operation; the rehuffle operation in the torage area; and o on. A a reult, the uncertaintie in the ytem become a huge challenge to the terminal daily operation. In uch a tochatic woring environment, deciion have to be made without complete nowledge of future event, oppoite to the tatic problem decribed in the lat two ection. Compared with the off-line dipatching rule, the on-line dipatching method i more appropriate in a highly dynamic woring environment where only limited information about future event i available. Than to the employment of advanced localization ytem and communication technologie, uch a Differential Global Poitioning Sytem (DGPS) and Radio Frequency Identification (RFID), the upervior can monitor the location and tatu of different equipment and container and communicate with operator and driver effectively (24). Thu, the fleet of vehicle can be dipatched following an on-line dipatching rule, uing the real-time proce and travel time. The on-line dipatching method are typically one-to-many aignment and all thee approache generally are claified into two categorie by Egbelu and Tanchoco (16). The firt category i a requet initiated dipatching rule that determine an 86

95 appropriate vehicle from a et of idle vehicle to match the tranportation requet. The econd category i vehicle-initiated dipatching rule that aign one requet from all available requet to the ingle vehicle. The on-line dipatching rule currently employed in NCT i the Longet Idle Vehicle (LIV) rule, according to the claification and definition in the wor of Egbelu and Tanchoco (16). Aume that there are two dummy depot at the ingle loop layout. The vehicle queue at depot A after dropping off container at the torage bloc and at depot B after dropping off container at the quay crane (FIGURE 14). Once there i a newly releaed tranportation requet, the firt two vehicle in the queue at depot A or B, which are alo the vehicle that remained idle the longet among all the idle vehicle, are dipatched to pic up the container from the QC ide or the YC ide. The main advantage of the rule i it eay application in the dynamic woring environment. However, there are two main hortcoming: One i that the rule doe not mae full ue of the vehicle reource. Every time a vehicle drop off a container at the QC or YC ide, it return to depot A or B, directly waiting for the next ta. Thu, the vehicle will not be dipatched to pic up the container on their way bac to the depot. The econd i that the rule doe not tae into conideration the imultaneou arrival of the two vehicle. For example, during the loading proce, although the two vehicle that are dipatched to pic up the two container may leave the depot almot at the ame time, it i till highly poible that they arrive at the QC ide at different time. 87

96 FIGURE 14 Current dipatching rule. To mae full ue of the vehicle capacity and mae account of the requirement of the tandem lift operation, thi ection propoe a heuritic method a the on-line dipatching rule. It i a priority-baed heuritic method with the objective of minimizing the maepan. The priority rule i not only eaily implemented, but alo flexible for practical application. The choice of attribute and the order in which they are compared can be adjuted according to the demand of the woring environment. Serie of numerical experiment are conducted to compare the performance of the propoed priority dipatching rule againt the LIV rule under different degree of tochaticity. The enitivitie of the approach to the QC cycle time and the vehicle average traveling peed are aeed in the enitivity analyi. 88