erli

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through EDI (electronic data interchange). Based on this informa-tion the container terminal generates predictive ship operationplan. After the container vessel arrives at the terminal, it rst waitsat anchor ground until its planned berth space becomes available.Then this vessel is berthed at the designated quay position, and

and QC scheduling can be inuenced by each other, hence thesetwo resources should be simultaneously considered while makingoperational plans. Because of the physical limitation of berth space(i.e. wharf length, water depth) and capacity limitation of quaycrane handling, well organized berth allocation and QC schedulingplans have great effect on terminals container turnover increase,hence the operational cost reduction for both terminal and vessels,and customer satisfaction improvement. As the service interfacebetween the terminal and vessels, high performance of quayside

* Corresponding author. Tel.: +86 13585930507; fax: +86 21 34206782.E-mail addresses: hanxl@sjtu.edu.cn (X.-l. Han), zhiqianglu@sjtu.edu.cn (Z.-q.

European Journal of Operational Research 207 (2010) 13271340

Contents lists availab

O

w.eLu), lfxi@sjtu.edu.cn (L.-f. Xi).nated container loading/unloading and transit tasks (Vis and deKoster, 2003). A well functional container terminal system requireseffective and collaborative decision making processes for resourceallocation and equipment scheduling. Typical decision problemsinclude ship planning process related to task arrangement andcoordination of arriving vessels, storage and stacking logistics inyard, and transport optimization of container transshipmentequipment (Steenken et al., 2004).

When container vessels are sailing to the destination port, theyperiodically update their ETA (estimated time of arrival) to theport, and the terminal acquires information about container load-ing/unloading tasks from vessel company and cargo agents

terminates and this vessel leaves the quayside, its occupied berthsegment is then released and becomes available again. Fig. 1 is asketch of 3 berthed vessels and 6 serving quay cranes at a containerterminal with 3 berths.

The above process is a brief description of quayside operation,and it follows berth and QC plans, the arrangement of which areamong the rst scheduling problems encountered by terminaloperation planners. Generally, berth scheduling is to determinethe berthing locations for arriving vessels in planning time horizon,along with sequences and time windows for their loading/unload-ing tasks; and QC scheduling is to determine each available QCsprocessing vessels and related processing time windows. The berth1. Introduction

Container transportation has beelogistics, and seaport container termnodes of international transportatiofor a container terminal due to itscompetition with others. A contaispace resources such as quay berthshipment equipment such as cranes0377-2217/$ - see front matter 2010 Elsevier B.V. Adoi:10.1016/j.ejor.2010.07.018mainstream of modernare considered as keyh performance is vitalapitalization and ercerminal system utilizesards, as well as trans-ucks to perform desig-

some pre-scheduled quay cranes are assigned to serve this vesseland start processing its container transshipment tasks. During itsprocessing, the vessels position is usually xed, because its reposi-tioning is time consuming, which involves unfasten of mooringcables and the help from a tugboat. But quay cranes can movearound along the guide rail as long as they do not interfere witheach other, to perform handling tasks of containers located at dif-ferent holds of the vessel, or at different vessels. When all con-tainer loading/unloading tasks of a vessel are nished, the serviceUncertainty 2010 Elsevier B.V. All rights reserved.Production, Manufacturing and Logistics

A proactive approach for simultaneous bproblem with stochastic arrival and hand

Xiao-le Han *, Zhi-qiang Lu, Li-feng XiDepartment of Industrial Engineering and Management, Shanghai Jiao Tong University,

a r t i c l e i n f o

Article history:Received 18 June 2009Accepted 17 July 2010Available online 23 July 2010

Keywords:Genetic AlgorithmStochastic simulationBerth and quay crane scheduling

a b s t r a c t

For a container terminal syimprovement of both operquay crane scheduling probtainer handling time. The bpriorities. QCs are allowedsels, adding more exibiliand a simulation based Gand QC schedule proactiveoped algorithm under unc

European Journal of

journal homepage: wwll rights reserved.th and quay crane schedulingng time

Dong Chuan Rd, Shanghai 200240, China

m, efcient berth and quay crane (QC) schedules have great impact on then efciency and customer satisfaction. In this paper we address berth ands in a simultaneous way, with uncertainties of vessel arrival time and con-s are of discrete type and vessels arrive dynamically with different serviceove to other berths before nishing processing on currently assigned ves-the terminal system. A mixed integer programming model is proposed,

tic Algorithm (GA) search procedure is applied to generate robust berthComputational experiment shows the satised performance of our devel-inty.

le at ScienceDirect

perational Research

lsevier .com/locate /e jor

tion 6 concludes this paper.

serv

erat2. Literature review

Many research works have been reported in the literature oneither berth or QC scheduling problem. The berth scheduling prob-lem can be divided into two categories: discrete and continuous,according to different indexing methods to locate the berthed ves-sels. In discrete situation, the wharf is divided into several sepa-rated segments, and no vessels berthing location can stride overadjacent segments (Imai et al., 2001; Wang and Lim, 2007). Inthe continuous situation, there is no partition of the wharf and ves-sels can be berthed anywhere along the wharf, as long as enoughspace exists (Kim and Moon, 2003; Imai et al., 2005). For the QCscheduling, the most important constraint is that quay cranes allmove along a shared guide rail, which forbids them from gettingacross each other in position (Lee et al., 2008; Tavakkoli-Moghad-dam et al., 2009). Bierwirth and Meisel (2010) provided a compre-hensiveness survey on both berth allocation and quay cranescheduling problems.

The amount of researches on simultaneous berth and QCscheduling problem is relatively small. Bierwirth and Meisel(2010) also provided a profound classication scheme of theseoperation also has direct or indirect impacts on the efciency ofother operational sections such as yard, yard cranes and transship-ment trucks, etc.

The real-time execution of operational schedules at containerterminal is affected by different kinds of uncertainties lying in ves-sel arrival time, task handling time, equipment reliability, con-tainer information inaccuracy, weather variability, etc. Like themachine scheduling in production, these uncertainties will causeextra cost and degrade the performance of the original schedule.One way to deal with these uncertainties is to generate a perturba-tion-insensitive robust schedule by taking into account uncertain-ties while making plans. Other ways include revision orrescheduling when uncertainty is realized, called the predictive-reactive approach (Aytug et al., 2005).

Based on practical container terminal operation, we look intothe simultaneous berth and quay crane scheduling problem, inwhich each vessels arrival time and handling time are stochasticdistributed. This paper is organized as follows. The next sectionprovides a literature review. The mathematic formulation of ourproblem is given in Section 3. Section 4 proposes a solution proce-dure, and numerical experiments are presented in Section 5. Sec-

Fig. 1. Quay cranes

1328 X.-l. Han et al. / European Journal of Opresearches based on two kinds of integration concepts: deep inte-gration and functional integration. Most researches of integratedberth and QC scheduling problem focus on continuous berth sit-uation. Park and Kim (2003) rst proposed a scheduling methodfor berth and quay cranes under continuous berth situation. Bothtime and space coordinates were discretized to formulate the MIPmodel, and a two phased solution procedure was adopted. In rstphase berth allocation and rough quay crane allocation wasdetermined, then in second phase detailed crane schedulingwas generated considering minimal setups times. Meisel andBierwirth (2009) investigated a similar problem with the rstphase problem in Park and Kim (2003), assuming that when mul-ti cranes are serving the same vessel, each cranes productivitywas decreased by interference with each other. They appliedtwo meta-heuristics, Squeaky Wheel Optimization and TabuSearch, respectively to alter the vessel priority list, and proposeda heuristic for searching better solutions under a given prioritylist. In these two studies under continuous berth situation, quaycranes can be moved to other vessel before its current vessel n-ishes processing, however such movement related time was notconsidered when evaluating vessel processing time. Zhang et al.(2010) considered the coverage ranges for quay cranes whenaddressing the simultaneous berth and quay crane schedulingproblem under continuous berth situation, and applied a sub-gra-dient optimization algorithm based on Lagrangian relaxation tosearch for near-optimal solutions. By limiting the quay craneadjustments, each decomposed sub-problem was solved by apolynomial-time enumeration procedure. Imai et al. (2008) con-sidered berth and QC simultaneously under discrete berth situa-tion. The authors used GA to generate the berth allocation ofvessels, and then proposed a heuristic to schedule crane transfer-ring. The feasibility of berth schedule was checked by convertingthe crane transferring problem into a network ow problem, andcrane tasks were rescheduled if necessary. Different from the twostudies under continuous berth situation, the authors assumedthat quay crane cannot move from one berth to another via otherberths if the other berths were engaged in vessel task operation,and the assigned QC amount for each vessel was pre-determined,hence the handling time of a vessel remained unchanged if moreQCs than needed were assigned to it. Liang et al. (2009) investi-gated a similar problem with Imai et al. (2008), and a combinedGenetic Algorithm with heuristic was proposed, in which a chro-mosome included both berth and quay crane assignment infor-mation. However the quay crane moving time was notconsidered and detailed algorithm for quay crane movementscheduling was not provided.

No impact from uncertainty on berth and quay crane schedulingis considered in these studies, which is quite important for sched-uling of complex and variable systems such as container terminal.However there exist many studies on uncertainty based schedul-ing, with applications in other different areas. Singer (2000) stud-ied a job shop scheduling problem with random processing timeand applied it in production planning. Acar et al. (2009) proposeda generalized MIP formulation with iterative optimization-simula-tion procedure and applied it in facility location problem. Morestudies were reviewed in Sahinidis (2004) and Aytug et al.(2005). In this paper, we address the simultaneous berth and quay

ing berthed vessels.ional Research 207 (2010) 13271340crane scheduling problem in which QC operation on one vessel canbe interrupted and resumed by other QCs later. Uncertainty factorssuch as stochastic vessel arrival and container handling time areconsidered.

3. Problem description and formulation

In our problem the berth is of discrete type, i.e. the wharf isdivided into discrete segment, and each vessel can berth into anysegment as long as the space constraints are satised, but cannotbe berthed across two successive segments. The vessels are arrived

eratdynamically, i.e. their arrival time can be after the planning mo-ment. Arriving vessels have different priority levels, representingrelative customer importance. All berths and QCs are available atthe beginning of scheduling time horizon. In our problem we makeplanning decisions on each vessels berth position, service se-quence and assigned QC amount. We assume when a new vesselberths, QCs can be reassigned among berths if its necessary to pro-vide enough QCs as planned for all berthed vessels. This meanseach QC can interrupt its service operation to one vessel and movealong the guide rail to start or resume processing another vessel,under the condition that after the movement, the assigned amountof QCs for each berthed vessel remains unchanged. In this way thevessels are not assigned with specic QCs but the amount of QCs.This assumption may cause more setup time because more fre-quent changes of QC positions, but brings into the system moreexibility. Instead of waiting until another vessel nishes opera-tion (which may be quite a long time), a newly arrived vessel canstart loading/unloading operation shortly after its arrived, as longas there are enough QCs to serve all berthed vessels.

Fig. 2 is a Gantt chart of 7 vessels processed at 4 berths. Eachrectangle represents a corresponding vessels processing startingtime and ending time, and is labeled with the vessels ID and as-signed quay crane amount. For instance the rectangle in berth-2 la-beled with 6(4) indicates vessel-6 will be moored at this berth,and 4 quay cranes will be assigned to handle its containers. Sup-pose there are 6 available QCs along the quayside. When vessel-3berths and starts processing at time t1, because there are alreadyenough QCs at berth-3 (2 QCs have just nished their tasks on ves-sel-1), thus no QC reassignment is required. However when vessel-6 berths at time t2, there willnot be enough QCs at berth-2 becauseall 6 QCs are busy or idle at other berths, if no QC moves amongberths from time t1 to t2. Hence QC reassignment among berthsand additional setup time for vessel-7 and vessel-2 are requiredat time t2. In Gantt chart the two shadowed segments representsthe setup time for vessel-7 and vessel-2 due to QC reassignment.

According to the classication scheme in Bierwirth and Meisel

Fig. 2. Example of Gantt chart for vessel processing.

X.-l. Han et al. / European Journal of Op(2010), our problem can be viewed as BAP; QCAPnumber QCAPspecific. We use the following notation to representparameters and variables that are involved in this problem:

Parametersi 2 I f1; . . . ;eIg set of berthsj 2 J f1; . . . ;eJg set of vesselsk 2 K f1; . . . ;eJg set of processing sequences of vessels at any

berthl 2 L f1; . . . ; eLg set of QCsh 2 H f1; . . . ; eHg set of service priorities of vesselsjh 2 Jh 1; . . . ; eJhn o subset of V, vessels with priority level hWi water depth of berth iQi wharf length of berth iDj tonnage of vessel j, including vertical safety distanceTj length of vessel j, including horizontal safety distanceAj arrival time of vessel j, which is a stochastic parameterwith probability density function uAj

Cj task handling time of vessel j by single QC, which is also astochastic parameter with probability density function uCj

Ej due time of vessel jBj max QC amount that can be assigned to vessel jbh weight value for vessels with priority hP setup time for QC reassignmentM number large enough

Decision variablesxijk = 1, if vessel j is processed at berth i as the kth vessel; = 0,

otherwisetijk processing starting time of vessel j at berth i as the kth ves-

selcijk processing time of vessel j at berth i as the kth vesselbijk assigned amount of QCs of vessel j at berth i as the kth ves-

selv jj0 = 1, if vessel j starts operation when vessel j0 is under pro-

cessing; = 0, otherwiserjj0 = 1, if vessel j starts operation after vessel j

0 starts process-ing; = 0, otherwise

sjj0 = 1, if vessel j starts operation before vessel j0 ends process-

ing; = 0, otherwiseej = 1, if vessel j is delayed by its due date; = 0, otherwisenji quay crane amount at berth i when vessel j entersujj0 = 1, if vessel j is the rst one to start operation after vessel

j0 starts operation; = 0, otherwisewjj0 = 1, if vessel j and j

0 start operation at the same time; = 0,otherwise

zj = 1, if reassignment of quay cranes among berths is neededwhen vessel j enters; = 0, otherwise

Given the two parameters Aj and Cj taking stochastic values, ourobjective is to proactively plan a robust berth and QC schedulewhich has statistically good performance under these uncertaintieswithout rescheduling. Survey and analysis of actual terminaloperation data show that under most ordinarily circumstances,these parameters are normal distributed, i.e. Aj NlAj ;rAj Cj NlCj ;rCj . Under some extreme circumstances like bad weathercondition or major failure of equipment, Aj or Cj uctuates so muchthat rescheduling is usually required, which induces a differentproblem from ours. All decision variables except xijk and bijk willbe stochastic variables because they are related to these stochasticparameters. Our problem is formulated as the following model:

min Z Ef rf 1s:t: f

Xi

Xj

Xk

tijk cijk Ajxijk

Xh

Xi

Xj2Vh

Xk

bhxijktijk cijk Ejej; 2Xi

Xk

xijk 1 8j; 3Xj

xijk 6 1 8i; k; 4Xj

xijk1 PXj

xijk 8i; kP 2; 5

Aj Mxijk 1 6 tijk 6 Mxijk 8i; j; k; 6Mxijk 1 < cijk 6 Mxijk 8i; j; k; 7Mxijk 1 < bijk 6 xijkBj 8i; j; k; 8tijk

Xj0tij0k1cij0 k1 P Mxijk 1 8i; j; k; 9

ional Research 207 (2010) 13271340 1329Xi

Xk

Wi Djxijk P 0 8j; 10

Xi

Xk

Qi Tjxijk P 0 8j; 11

rjj0 1M 6Xi

Xk

tijk Xi

Xk

tij0k < rjj0M 8j j0; 12

sjj0 1M