Robust berth scheduling at marine container terminalsvia hierarchical optimization

Mihalis Golias a,n, Isabel Portal b, Dinçer Konur c, Evangelos Kaisar d, Georgios Kolomvos e

a Department of Civil Engineering and Intermodal Freight Transportation Institute, University of Memphis, United Statesb Department of Ocean and Mechanical Engineering, Florida Atlantic University, United Statesc Department of Engineering Management and Systems Engineering, Missouri University of Science and Technology, United Statesd Department of Civil, Environmental and Geomatics Engineering, Florida Atlantic University, United Statese Industrial Engineering and Optimization Laboratory, Kathikas Institute of Research and Technology, Cyprus

a r t i c l e i n f o

Available online 9 August 2013

Keywords:Berth schedulingContainer terminalsBi-levelBi-objectiveMetaheuristics

a b s t r a c t

In this paper, we present a mathematical model and a solution approach for the discrete berth schedulingproblem, where vessel arrival and handling times are not known with certainty. The proposed modelprovides a robust berth schedule by minimizing the average and the range of the total service timesrequired for serving all vessels at a marine container terminal. Particularly, a bi-objective optimizationproblem is formulated such that each of the two objective functions contains another optimizationproblem in its definition. A heuristic algorithm is proposed to solve the resulting robust berth schedulingproblem. Simulation is utilized to evaluate the proposed berth scheduling policy as well as to compareit to three vessel service policies usually adopted in practice for scheduling under uncertainty.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Marine container terminals are essential components of globalsupply chains and terminal operators face many challenges on day today operations to remain efficient and competitive. Most of thesechallenges are due to the interactions between different operationsand facilities within the terminals. For instance, berth scheduling withrespect to vessel characteristics considering the storage, loading, andunloading operations at the yard side is a daily challenge for terminaloperators. Another challenge comes from assigning vessels to theavailable berths depending on different service policies followed bythe terminal operator [1,2]. Terminal operators should overcome thesechallenges to improve their performance measures and stay compe-titive. Specifically, there is a growing industry interest to expand themetrics in reliability and understand the gap in container delivery andliner schedule reliability.1 It is, therefore, crucial to evaluate the causesand the effects of shipment delays.

To stay competitive, terminal operators should develop accurateand reliable berth schedules in order to avoid its customers' ship-ment delays. Berth scheduling refers to the allocation of a specificvessel to a particular physical location within the port for loading/unloading processes. This paper concentrates in the development of

a berth schedule that explicitly accounts for the uncertainties invessel arrivals and handling times. Even though port operators mayhave an estimated vessel arrival time window, it is difficult to knowthe exact arrival time in advance (e.g., delay due to weather, delay atport of origin). In addition, due to a number of operational factors(e.g., quay crane breakdowns, yard congestion, changes to stowageplan, etc.) terminal operators can usually only estimate an upper anda lower bound on the vessels' handling times.

A significant amount of research has been conducted to analyzeand improve berth scheduling policies (we refer to [2] for anexcellent and recent literature review and classification on seasideoperations at marine container terminals). The research studies onberth scheduling could be classified considering four mainassumptions on the inherent characteristics of the problem [1,2].The first assumption is on the definition of the berth spaceas discrete or continuous space. In case of discrete space, the wharfis divided into a specific number of berths. On the other hand,continuous space defines the entire wharf as the berth space. Wenote that a third case exist (hybrid) where the wharf is discretizedin berths but multiple vessels can be served at each berth simul-taneously [2]. The second assumption is on the dynamic or staticnature of vessel arrivals. Particularly, while static vessel arrivalsrefer to the case where all vessels are at the port when the scheduleis developed, the dynamic vessel arrivals formulation assumes thatarrival times are known within a time window. The third assump-tion refers to deterministic versus stochastic nature of handlingtimes. While handling times are assumed to be known in determi-nistic handling time formulations, handling times are assumed to

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/caor

Computers & Operations Research

0305-0548/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.cor.2013.07.018

n Corresponding author. Tel.: +1 901 678 3048; fax: +1 901 678 3026.E-mail addresses: mihalisdgolias@gmail.com,

mgkolias@memphis.edu (M. Golias).1 American Shipper: New report compares liner, container delivery reliability.

[Internet]. [Cited: 20 July 2012]. Available from http://www.americanshipper.com/.

Computers & Operations Research 41 (2014) 412–422

be random variables in stochastic handling time formulations. Thefourth assumption refers to performance measurements or touncertainty of the problem parameters; i.e., handling and arrivaltimes. Most of the studies in the current literature focus on and varyin their assumptions on the first two characteristics and limitedresearch has been published on analyses of the latter twocharacteristics.

In the setting of this paper, discrete berth space and uncertainvessel arrivals and handling times are assumed. We propose amathematical formulation (extending the work by Konur andGolias [3]) that simultaneously minimizes the average and rangeof the total service time required to serve all the vessels at theterminals. The problem is initially formulated as a bi-objectiveoptimization problem that contains two optimization problems inthe definitions of each of the two objective functions. To overcomethis complexity, the problem is decomposed into a bi-objective bi-level optimization problem. The revised formulation simplifies theproblem and provides means to address both objectives in isola-tion. To solve the resulting problem, we propose a heuristic appr-oach that combines exact and heuristic solution methods. Theproposed scheduling strategy is compared to three commonlyused scheduling strategies under arrival and handling time uncer-tainty: First Come First Serve Early Start (FCFS-S), First Come FirstServe Early Finish (FCFS-F), and Expected Arrival and HandlingTime Scheduling (EAHTS).

To the best knowledge of the authors, this study is the firstin analyzing robustness in case of uncertain vessel arrival andhandling times for the berth scheduling. The rest of the paper isstructured as follows. Section 2 reviews the berth schedulingliterature. Section 3 explains the proposed robust schedulingapproach and provides the mathematical formulations. The pro-posed solution algorithm is explained in Section 4. Section 5present results from a number of numerical examples and com-pares the proposed approach to three commonly used berthscheduling policies under arrival and handling time uncertainty.Summary of contributions, results and findings, and possible futureresearch directions are noted in Section 6.

2. Literature review

A significant amount of research has been recently conductedon the berth scheduling problem (BSP) but only limited researchhas been published that deals with uncertainty when evaluatingberth scheduling strategies. Golias [4] formulates a bi-objectivemixed integer programming problem to simultaneously maximizethe berth throughput and the reliability of a berth schedule. In thisBSP, the vessel handling times are defined as stochastic variablesand the author uses a combination of an exact algorithm andgenetic algorithm based heuristics to solve the BSP with stochasticvessel handling times. Moorthy and Teo [5] address a BSP withcontinuous berth space and dynamic vessel arrivals to developberth templates. The authors minimize the vessel waiting timeand the cost of vessel transshipments. Their approach (as statedby the authors) is relevant only when a substantial number ofvessels arrive periodically. Golias et al. [6] present a conceptualformulation for the discrete space dynamic vessel arrival BSP,where both vessel arrival and handling times are considered asstochastic variables. The authors present and compare the resultsof the following four heuristics based solution approaches: MarkovChain Monte Carlo based heuristic, online stochastic optimizationbased heuristic, deterministic solution based heuristic, and acombination of Monte Carlo with online stochastic optimizationbased heuristic. Zhou et al. [7] and Zhou and Kang [8] proposesimilar models dealing with uncertainty in vessel handling andarrival times by introducing probabilistic constraints. We note that

formulations by Zhou et al. [7] and Zhou and Kang [8] constrainthe vessel waiting times, which may lead to infeasibility (i.e., strictwaiting time limits) or low quality solutions (i.e., high waitingtime limits). Furthermore, their models are highly non-linear andassume normal distributions of the vessel arrival and handlingtimes; whereas, use of Poisson, Uniform, truncated Normal, Gamma,or Erlang distributions are noted to represent vessel arrival andhandling times in the literature [9,10].

Du et al. [11] study BSP with continuous space and proposea reactive feedback procedure to develop robust berth schedulesaddressing the uncertainty in vessel delays. The authors measurescheduling robustness with the cost of berthing a vessel at a non-preferred berth and the cost of delayed berthing and departure.These measures of robustness, nevertheless, may be conflictingas delayed berthing may result in early departures or assignmentto a non-preferred berth may result in reduced waiting time andearly departures. Gao et al. [12] address uncertainties in vesseldelays and out-of-schedule berthing by proposing two types ofstrategies to develop a berth schedule: a proactive strategy isdeveloped with a feedback procedure in case of vessel delays and areactive strategy with a reassignment rule is proposed for out-of-schedule vessels. Xu et al. [13] propose a BSP formulationthat models uncertainty in vessel delays and handling times byintroducing the concept of buffers for the delays. This formulation,however, assumes continuous berth space, and; therefore, cannotbe applied to the discrete case. In a recent study, Zhen et al. [14]apply the scenario-based modeling to berth scheduling. Theauthors introduce a number of scenarios for the vessel arrivalsand operation times and develop a two-stage decision model,which is solved through a meta-heuristic. Han et al. [15] representthe uncertainty in vessel arrivals and handling times throughprobability density functions. When this type of an approach isadopted, it is common that simulation and heuristic proceduresare used as solution methods.

The framework presented in this paper extends the currentliterature on BSPs by developing a proactive robust berth schedul-ing strategy that does not require knowledge or use of probabilitydistributions for the vessel arrival and handling times. Amongthe common methods to introduce uncertainty in mathematicalprogramming models are modeling through: (a) a scenario space,(b) a probability distribution, or (c) sets of upper and lower bounds.Modeling uncertainty through a scenario space is usually appro-priate if the uncertainties inherent in the processes have beenpreviously observed for relatively long periods of time and ahistorical set of data has been collected. This scenario-drivenformulation approach usually leads to two-stage and multistagemodels of stochastic programming.

Modeling uncertainty in mathematical problems through intro-duction of upper and lower bounds on the variables has beenextensively studied by Ben Tal and Nemirovski [16] and is adoptedin this paper. This type of approach falls under the area of robustoptimization. Solution techniques for these types of problemsusually lead to the semi-definite and conic optimization models,which are generally non-linear. The term “robust optimization”has also been attributed to variable dissimilar formulations,all converging to the common aim of producing solutions whichwill not be catastrophic, in terms of costs, following the realizationof the random variables. Next we present the motivation for thisstudy and the problem formulation.

3. Model formulation

As noted previously, robustness of a berth schedule is crucialfor the overall performance of a container terminal [13]. A berthschedule can be defined to be robust in case the total time

M. Golias et al. / Computers & Operations Research 41 (2014) 412–422 413

required to serve vessels does not vary with the inherent uncer-tainties of vessel arrival and handling times. Therefore, the rangeof the total vessel service time of a given berth schedule can beused to quantify the robustness of that schedule. The range of totalvessel service time is determined by the difference between theworst and best performances of a given berth schedule. Therefore,in modeling the range of total service time, a bi-level formulationis required (particularly, to determine the worst and the bestperformances of the given schedule). Nevertheless, a schedulewith efficient robustness can poorly perform on average, that is,while the range of the total vessel service time is small, averagetotal vessel service time can be high. To overcome this issue, theberth scheduling problem is formulated as a bi-objective optimi-zation problem. That is, the modeling approach detailed next seeksa robust berth schedule with good average performance. Prior tothe model formulation, we next discuss the motivation andpresent the necessary notation.

Consider a marine container terminal with n berths and mvessels that request service (loading/unloading of the containers).Let i and j be the index of the berths and vessels, respectively, suchthat iA I; I¼ f1;…;ng andjA J; J ¼ f1;…;mg. A solution to the berthscheduling problemwill consist of vessel to berth assignments andthe order of service for each vessel (at its assigned berth).To define a berth schedule, we first define the following variables:

xij ¼ 1 if vessel j is assigned for service at berth i and zerootherwise,yab ¼ 1 if xia ¼ xib ¼ 1 and b is the immediate successor of a andzero otherwise,f j ¼ 1 if vessel j is the first vessel to be served at its assignedberth and zero otherwise,lj ¼ 1 if vessel j is the last vessel to be served at its assignedberth and zero otherwise.

A berth schedule can be defined by S(X,Y), where X is a n�mbinary matrix that defines the vessel to berth assignments and Y isa n�m binary matrix that defines the service orders of the vessels. IfS is known, the start and finish times of each vessel (stj and ftj,respectively) can be determined (assuming that the handling andarrival times for each vessel are known in advance). In this paper, weassume that each vessel has a unique handling time at each berth,which depends on a number of factors such as relative location ofstorage area for vessels' unloaded containers, number of quay cranes,and internal transport vehicles assigned to the vessel. Furthermore,it is assumed that the vessel handling times are randomvariables withupper and lower bounds. Similarly, we assume that the arrival time foreach vessel is a random variable with upper and lower boundsbecause port operators usually receive a time window for vesselarrivals. The lower and upper bounds on vessel arrivals and handlingtimes define the minimum and maximum values for vessel arrivalsand handling times, respectively, and these values are known to theport operator. Therefore, the inherent uncertainty of berth schedulingthat stems from these two variables are captured in this paper.

Let ½Alj;A

uj � be the lower and upper bounds of vessel j's arrival time

and ½clij; cuij� be the lower and upper bounds of the handling time of

vessel j at berth i. Furthermore, let A denote a m-vector of vessel

arrival times such thatAjAfAlj;A

uj g; 8 jA J and C denote a n�m-matrix

of vessel handling times such thatcjAfclij; cuijg; 8 iA I; jA J. The berth

scheduling problem that minimizes the average total vessel servicetime can then be defined as follows:

f 1 : minX;Y

12

maxC;A

∑jA J

ðstj�AjÞ þ ∑iA I;jA J

cijxij

( )þmin

C;A∑jA J

ðstj�AjÞ þ ∑iA I;jA J

cijxij

( ) !" #

ð1Þ

subject to.:

∑iA I

xij ¼ 1; 8 jA J ð2Þ

f b þ ∑aabA J

yab ¼ 1; 8bA J ð3Þ

la þ ∑aabA J

yab ¼ 1; 8aA J ð4Þ

f a þ f br3�xia�xib ¼ 1; 8 iA J; a; bA J; aab ð5Þ

la þ lbr3�xia�xib ¼ 1; 8 iA J; a; bA J; aab ð6Þ

yab�1rxia�xibr1�yab; 8 iA J; a; bA J; aab ð7Þ

stjZAj; 8 jA J ð8Þ

stbZsta þ ∑iA I

ciaxia�Mð1�yabÞ; 8a; bA J; aab ð9Þ

AljrAjrAu

j ; 8 jA J ð10Þ

clijrcijrcuij; 8 iA I; jA J ð11Þ

The objective function (1) minimizes the average total service timefor all the vessels, which is defined as the arithmetic average ofthe maximum and minimum total service times possible. Constraintset (2) ensures that each vessel is served once and constraintset (3) ensures that each vessel is either served first or preceded byanother vessel. In a similar manner, constraint set (4) guarantees thateach vessel is either last or served before another vessel. Constraintsets (5) and (6) ensure that only one vessel can be served as thefirst and the last vessel at each berth, respectively. Constraintset (7) enforces that a vessel can be served after another vessel onlyif both vessels are served at the same berth. Constraint set (8) restrictsthat the vessel service start times are greater than the vessels' arrivaltimes. Constraint set (9) estimates the start time of service for eachvessel. Constraint sets (10) and (11) define the values of the arrivaland handling times between their respective bounds, respectively.

Scheduling solely based on the average total service time mightlead to schedules that are not robust, that is, high range of totalservice times as C and A vary. To account for the robustness of theberth schedule, we introduce a second objective that minimizesthe range of the total service time. In particular, given a scheduleS(X,Y), we minimize the following equation:

f 2 : minX;Y

maxC;A

∑jA J

ðstj�AjÞ þ ∑iA I;jA J

cijxij

( )�min

C;A∑jA J

ðstj�AjÞ þ ∑iA I;jA J

cijxij

( ) !" #

ð12Þwhere the first and second components are the maximum andminimum total service times possible, respectively.

Both objective functions have two optimization problems in theirdefinition (i.e., maximization and minimization of total service times).To overcome this issue, in what follows, we reformulate the problemas a bi-objective bi-level optimization problem. To reformulate the

problem, let ½Cmax;Amax�ðSÞ and ½Cmin;Amin�ðSÞbe the handling andarrival time instances (over their respective ranges) that maximize andminimize the total service time of schedule S(X,Y). To determine

½Cmax;Amax�ðSÞ and ½Cmin;Amin�ðSÞ, one needs to solve the followingoptimization problems:

f 3 : max=minC;A

∑jA J

ðstj�AjÞ þ ∑iA I;jA J

cijxij

( )ð13Þ

subject to:

clijrcijrcuij; 8 iA I; jA J ð14Þ

M. Golias et al. / Computers & Operations Research 41 (2014) 412–422414

AljrAjrAu

j ; 8 jA J ð15Þ

∑aA J:aa j

stayaj þ ∑iA I;aA J:aa j

ciaxiayaj�AjrMð1�zjÞ; 8 jA J ð16Þ

Aj� ∑aA J:aa j

stayaj þ ∑iA I;aA J:aa j

ciaxiayajrMzj; 8 jA J ð17Þ

stj� ∑aA J:aa j

stayaj þ ∑iA I;aA J:aa j

ciaxiayajZ0; 8 jA J ð18Þ

stj�AjZ0; 8 jA J ð19Þ

Aj�stj þMð1�zjÞZ0; 8 jA J ð20Þ

zjAf0;1g; 8 jA J ð21Þwhere zjAf0;1g; 8 jA J is an auxiliary variable used to bound the starttime of the vessels. Note that when the objective is maximization(minimization) in (13), f3 determines the worst (best) performance ofthe given berth schedule. Constraints (14) and (15) ensure that thehandling and arrival times are within the given time windows,respectively. Constraints (16)–(20) define the service start times ofthe vessels. Particularly, they assure that the service start time ofa vessel is the earliest of the finish time of its immediate predecessoror its arrival time.

The original problem can then be reformulated as follows:

f 1 : minX;Y

12

∑jA J

ðstmaxj �Amax

j Þ þ ∑iA I;jA J

cmaxij xij

( ) "

þ ∑jA J

ðstminj �Amin

j Þ þ ∑iA I;jA J

cminij xij

( )!#

f 2 : minX;Y

∑jA J

ðstmaxj �Amax

j Þ þ ∑iA I;jA J

cmaxij xij�∑jA Jðstmin

j �Aminj Þ

�∑iA I;jA Jcminij xijX;Y

!

subject to: Eqs. (2)–(11)

ðCmax;AmaxÞ ¼ argmax ∑jA J

ðstj�AjÞ þ ∑iA I;jA J

cijxij

( ); subject to : Eqs: ð14Þ–ð21Þ

ðCmin;AminÞ ¼ argmin ∑jA J

ðstj�AjÞ þ ∑iA I;jA J

cijxij

( ); subject to : Eqs: ð14Þ–ð21Þ

A summary of the notation used throughout this section can beseen in Appendix. Next, we discuss the solution algorithm used tosolve the above problem.

4. Solution algorithm

Previous sections explain the importance of producing an efficientand robust berth schedule as well as the complexity of the robustberth scheduling problem. It is known that bi-level optimizationproblems are non-convex and hard to solve with exact solutionalgorithms. Therefore, we develop the following solution algorithm,which uses a genetic algorithm based heuristic approach to solve therobust berth scheduling problem. Particularly, the algorithm consistsof the following five steps: chromosome representation and initi-alization (Step 1), objective function evaluation (Step 2), Pareto Front(PF) selection (Step 3), crossover and mutation (Step 4), and termina-tion (Step 5). In Step 1, the problem is presented using integerchromosomes to apply genetic algorithm effectively. Step 2 explainshow to evaluate a given berth schedule in the form of a chromosome.As a bi-objective formulation is considered, in Step 3 of the solutionalgorithm, we focus on determining a set of Pareto efficient berth

schedule from a given set of schedules. Step 4 details the proceduresto create new berth schedules as chromosomes to find better Paretoefficient schedules. Finally, Step 5 discusses the termination criteriafor the solution algorithm. Each step of the solution algorithm isdescribed next.

4.1. Step 1: chromosome representation and initialization

In this paper, we use an integer chromosomal representation asillustrated with an example in Fig. 1 for a problem instance with2 berths and 6 vessels. In this example, vessels 1–3 are scheduled forservice at berth 1 and vessels 4–6 at berth 2. We initialize a populationof 100 chromosomes on a First Come First Served with early start(FCFS-S) policy and First Come First Served with early finish (FCFS-F)policy using expected arrival and handling times (assuming that bothparameters follow a uniform distribution within the time windowsprovided). That is, given particular arrival and handling time windowsof the vessel arrivals and handling times, a chromosome thatrepresents the vessel to berth scheduling is created using FCFS-F andFCFS-S policies described in the following procedures.

4.1.1. First come first served with early start (FCFS-S):

0: Setm as number of vessels, n as number of berths, andM as alarge number

1: Sort arrivals such Aj�1rAj; 8 jA J

2: Set Aj ¼ AljþAu

j

2

� �; Set cij ¼

clijþcuij2

� �; Set X ¼ zerosðm;nÞ;

Set f t ¼ zerosðn;1Þ3: For j¼ 1 : m4: Set Ar¼ Aj; Set St ¼M5: For i¼ 1 : n6: Set ws¼maxðf ti;ArÞ7: If wsoSt; Set b¼ i; Set St ¼ws8: Elseif ws¼ St and cijocb;j; Set b¼ i; Set St ¼ws9: End10: End11: Set Xj;b ¼ 1; Set Startj ¼ St; Set Finishj ¼ St þ cj;b; Setf tb ¼ Finishj

12: End13: Return X

4.1.2. First come first served with early finish (FCFS-F):

0: Setm as number of vessels, n as number of berths, andM as alarge number

1: Sort arrivals such Aj�1rAj; 8 jA J

2: Set Aj ¼ AljþAu

j

2

� �; Set cij ¼

clijþcuij2

� �; Set X ¼ zerosðm;nÞ;

Set f t ¼ zerosðn;1Þ3: For j¼ 1 : m4: Set Ar¼ Aj; Set Et ¼M5: For i¼ 1 : n6: Set st ¼maxðf ti;ArÞ; Set wf ¼ st þ cij7: If wf oEt; Set b¼ i; Set Et ¼wf8: Elseif wf ¼ Et and cijocb;j; Set b¼ i; Set Et ¼wf

Berth 1 2

Vessel 1 2 3 4 5 6

Fig. 1. Chromosome representation.

M. Golias et al. / Computers & Operations Research 41 (2014) 412–422 415

9: End10: End11: Set Xj;b ¼ 1; Set Finishj ¼ Et; Set Startj ¼ Et�cj;b;Set f tb ¼ Finishj

12: End13: Return X

4.2. Step 2: objective function evaluation

Given a chromosome (i.e., a schedule S(X,Y)), we calculate thevalues of both objective functions as follows. To obtain ½Cmax;

Amax�ðSÞand ½Cmin;Amin�ðSÞ, we first solve f 3 defined in Eqs. (13)–(21). Note that f 3 is separable at each berth if S(X,Y) is given. Once wehave the arrival and handling times (½Aj;Cij�; 8 iA I; jA J) for a chro-mosome that maximize and minimize total service time of scheduleS(X,Y), we can easily calculate the values of both objective functions.There are two approaches that could be used to solve f 3. The firstapproach is to solve f 3 to optimality (i.e., CPLEX can be used for this)and the second approach involves the use of heuristics. The latterapproach is selected in this paper due to computational timerequired by CPLEX to solve the lower level problems (and lowerlevel problems are required to be solved repeatedly). In particular,two separate heuristics are proposed to determine ½Cmax;Amax�ðSÞand½Cmin;Amin�ðSÞ. Before we proceed with the description of theseheuristics, we present the following proposition that will be utilizedwithin the proposed heuristic approaches for f 3.

Proposition. For any (X,Y,A) the maximum and minimum valuesof f 3 occur when cij ¼ cuij; 8 jA J; iA I and cij ¼ clij; 8 jA J; iA I,respectively.

Proof. Increasing/decreasing the value of C increases/decreases(linearly) the value of f3 and vessel start times (Eqs. (16)–(20)).Thus, the maximum and minimum value of f3 occurs when cij ¼ cuij;8 jA J; iA Iandcij ¼ clij; 8 jA J; iA I, respectively. □

4.2.1. Minimum search heuristic (MISH)MISH is used to determine ½Cmin;Amin�ðSÞ.Given a schedule S(X,

Y), arrival and handling times for each vessel at its assigned berthminimizing the total service time can be estimated using thefollowing procedure:

For each berth i

0: Set RBPi ¼ 0 and f to ¼ 01: For j¼ 1 : m2: Set cij ¼ clij3: If j¼ 1 orf tj�1rAl

j; set Aj ¼ Alj; set stj ¼ Aj;

set f tj ¼ stj þ cij4: Elseiff tj�14Au

j ; set Aj ¼ Auj ; set stj ¼ Aj; set f tj ¼ stj þ cij

5: ElseifAuj 4 f tj�14Al

j; set Aj ¼ f tj�1; set stj ¼ Aj;set f tj ¼ stj þ cij

6: End7: SetRBPij ¼ ðstj�AjÞ þ cij; set RBPi ¼ RBPi þ RBPij

8: End

9: Set Amin ¼ Aj; Set Cmin ¼ cij

10: Return Amin;Cmin.

4.2.2. Maximum search heuristics (MASH)MASH is used to determine½Cmax;Amax�ðSÞ. Similar to MISH,

given a schedule S(X,Y), arrival and handling times for each vessel

at its assigned berth maximizing the total service time can beestimated using the following procedure:

For each berth i

0: Set RBPi ¼ 0 and f to ¼ 01: For j¼ 1 : m2: Set cij ¼ cuj3: If j¼ 1 or f tj�1rAl

j; set Aj ¼ Auj ; set stj ¼ Aj;

set f tj ¼ stj þ cij4: Else; set Aj ¼ Al

j; set stj ¼ f tj�1; Set f tj ¼ stj þ cij5: End6: Set RBPij ¼ ðstj�AjÞ þ cij; set RBPi ¼ RBPi þ RBPij

7: End8: Set Amax ¼ Aj; set C

max ¼ cij9: Return Amax;Cmax.

4.3. Step 3: Pareto front (PF) selection

Given a generation of evaluated chromosomes, we next deter-mine the PF that will move on to the next generation. The PF refersto the set of non-dominated solutions, that is, these solutions arethe best solutions available that meet both objectives (minimizethe range and the average of total service time). A schedulebelongs to the PF if there is no other schedule that can improveat least one of the objectives without degradation of the otherobjective [17]. The initial PF is selected from the set of schedulesconsisting of the chromosomes of the current generation. Then,a random number of schedules are selected from the PF to performthe crossover and mutation operations. The PF is updated usingthe schedules within the new generation of chromosomes. Weinclude solutions of the previous PF to assure continuous improve-ment (until convergence). As we have a limited number of solu-tions, we use an exact enumeration method [3] to find the PF ofthe current generation. Note that the complexity of the algorithmis O(n2). The PF selection algorithm is described next.

Let LS represent the list of schedules such that (ATSls, RTSls)pair corresponds to the lsth schedule in LS, where ATSls is theaverage total service time for this schedule and RTSls is the rangeof total service time for this schedule. That is, ATSls and RTSlsrepresent the values of the first and the second objective function,respectively. Then, PF is obtained with the following procedure.

0: For ls¼ 1 : jLSj1: For k¼ ðlsþ 1Þ : jLSj2: If ASTlsoASTk and RSTlsoRSTk

3: Set LS : ¼ LS�fðASTk;RSTkÞg and go to step 14: Elseif ASTls4ASTk and RSTls4RSTk

5: Set LS : ¼ LS�fðASTls;RSTksÞg and go to step 06: End7: End8: End9: Return PF ¼ LS.

4.4. Step 4: crossover and mutation

To obtain a new generation of chromosomes, we performcrossover and mutation to the chromosomes in the current PF.We apply a single point crossover and four types of mutationsnamely swap, insert, invert, and scramble. Crossover is randomlyperformed between the chromosomes of the current PF until

M. Golias et al. / Computers & Operations Research 41 (2014) 412–422416

it produces a number of children chromosomes equal to thecurrent number of parents in the current PF (or a minimum of100). The four mutations are applied to every chromosomegenerating four mutated children per parent chromosome. Fig. 2illustrates the crossover and Fig. 3 illustrates the mutations for theexample shown in Fig. 1. Although the mutation operations arestraightforward, the crossover operations need further elabora-tion. As shown in Fig. 2, in the example crossover operation vessel1 replaced vessel 5 as the third vessel at berth 1. For the scheduleto remain feasible, vessel 5 was moved to berth 2 at the place ofvessel 1.

4.5. Step 5: termination

The algorithm is terminated if no improvement is observed inthe PF for 100 iterations or the CPU time exceeds 10 min. Fig. 4summarizes the solution algorithm in a flowchart.

5. Numerical examples

Problems used in the experiments are randomly generatedwithin a systematical approach. We develop 48 problem instances,where vessels' inter-arrival times vary and arrival time windowshave a range of 1–2 days. Also, vessels are served with varioushandling volumes (4 sets) at a multi-user container terminal (MUT)with varying number of berths (4 and 5 berths), with a planninghorizon of one week. Vessels' expected inter-arrival times aregenerated from an exponential distribution with means of 3, 4,and 5 h. Shorter inter-arrival times increase congestion at the MUT.Arrival time windows for each vessel are developed based on theuniform distributions over the arrival time window ranges shownin Table 1 (with the expected arrival time being the mean of thetime window). Vessel handling time volumes (time of loading andunloading) are developed based on uniform distributions for eachrange with four types of percentage handling volumes. We assumethat, for each vessel, there exists a berth with the minimumexpected handling time (preferred berth). The locations of the

preferred berths for the vessels are randomly selected. The mini-mum expected handling times of the vessels at the other berths aregenerated in relation to the handling time at the preferred berth byincreasing proportionally (10%) to the distance from the preferredberth. The minimum handling time of a vessel at the preferredberth is assumed to be a function of the handling volume, which is afunction of the resources allocated such as quay cranes and internaltransportation vehicles. In this case, we estimate three types ofhandling time volumes: 8–24 h for small vessels, 24–40 h formedium vessels, and 32–48 h for large vessels. The upper andlower bounds for the arrival windows are created using a uniformdistribution over the selected time window range. The upperbounds for the handling times are randomly generated usinga uniform distribution with a range of 120–150% of the expectedhandling times and the lower bounds for the handling times arerandomly generated using a uniform distribution with a range of50–80% of the expected handling times.

Each problem instance is solved with the proposed heuristic and aPF is obtained for each problem instance. Once these schedules areobtained, we simulate different vessel arrival and handling timecombinations to compare the schedules in the PF. We select the bestPF solutions based on the average simulated cost, which will bereferred as Pareto Front with Best Mean (PFBM). We compare theschedules from PFBM of the total service timewith two first come firstserved policies: (i) first come first served at the berth with the earliest

BerthVessel 2 4 5 0 0 0 1 3 6 0 0 0BerthVessel 2 4 5 0 0 0 1 3 6 0 0 0

BerthVessel 2 4 1 0 0 0 5 3 6 0 0 0

PARENT CHROMOSOMES

1 2

CHILDREN CHROMOSOME1 2

1 2

Fig. 2. Schematic illustration of crossover.

Vessel 2 4 5 0 0 0 1 3 6 0 0 0 2 4 5 0 0 0 1 3 6 0 0 0

Berth

Berth

Berth

BerthVesselBerth

VesselBerth

VesselBerth

VesselBerth

Vessel

Vessel

Vessel

2 3 4 5 0 0 1 6 0 0 0 0 4 3 1 0 0 0 6 5 2 0 0 0

2 4 5 0 0 0 1 3 6 0 0 0 2 4 5 0 0 0 1 3 6 0 0 0

3 4 5 0 0 0 1 2 6 0 0 0 5 4 2 0 0 0 6 3 1 0 0 0

1 2Before

INSERT Mutation

After

SWAP Mutation

Before1 2

1 2 1 2

1 21 2After

SCRAMBLE Mutation

Before1 2

After

INVERT Mutation

Before1 2

After

Fig. 3. Schematic illustrations of mutation.

Find arrival and handling times that maximize (MASH) and minimize

(MISH) total service time

Initialize chromosomes based on FCFS-S and FCFS-F policies

Join chromosomes from current generation and Pareto Front of

previous generation

Find Pareto Front

Algorithm Converged

End

Crossover and Mutation

No

Yes

Fig. 4. Flowchart of the solution algorithm.

M. Golias et al. / Computers & Operations Research 41 (2014) 412–422 417

start time (FCFS-S) and (ii) first come first served at the berth with theearliest finish time (FCFS-F). In addition, we compare the Paretoschedules to the schedule that minimized the total vessel service timegiven an expected handling and arrival time (EAHTS); where theexpected handling and arrival time is the mean arrival and meanservice time. The vessel arrival and handling times are simulatedbased on a triangular distribution as it can be used to representcontinuous asymmetric vessel arrival and handling times over a givenrange. Furthermore, assuming a triangular distribution is not restric-tive and can simply be used to represent different vessel arrival andhandling characteristics. In particular, we consider three cases for eachuncertain parameter, where the vessel arrives early, on time, and late,and the vessels' handling time is less, equal, or greater than itsexpected value. For each of the two uncertain parameters and eachof the three cases, we develop 48 scenarios with varying percentagesof vessels arriving early, on-time, late with handling times less, equalor greater than its expected value. For each scenario, we developed100 arrival and handling time realizations and obtain the berthschedules for the FCFS-S, FCFS-F and EAHTS. The latter scheduleis obtained using the heuristic developed by Golias [18].

Results of the simulation for all four strategies are shown in Table 2,where the average service time (from here on, service time is referredto as cost) per vessel for each dataset over the 24 scenarios and 100

arrival and handling time combinations are shown for problems with4 and 5 berths, respectively. Results are shown in an increasing orderof congestion based on the average cost for each vessel under PFBM.From the results of the simulation, it could be seen that as thecongestion levels increase in a container terminal, the PFBM strategyoutperforms the other strategies. For example, for the 4 berthsterminal case with high congestion (2.82 average cost per vesselshown in the last row/first cell of Table 2), PFBM outperforms the twoof the FCFS and EAHTS policies (3.35, 3.59. 3.18 average cost per vesselfor FCFS-F, FCFS-S, and EAHTS, respectively). On the other hand, whencongestion is low (0.79 average cost per vessel as shown in the firstrow of Table 2), the FCFS policies provide a lower average cost.Similarly, in the 5 berths terminal with high congestion (2.92 averagecost per vessel shown in the last row of Table 2), the PFBM schedulesprovide a lower average cost (2.92 days) when compared to FCFS-F(3.14 days), FCFS-S (3.40 days), and EAHTS (3.52 days) schedules. Therange for each of the strategies could be seen on the right part ofTable 2. The range provided by the PFBM strategy is lower than theother strategies. For instance, when high levels of congestion arepresent in a 4 berths terminal, the range of PFBM strategy is 0.28 dayswhile the other policies exceed a day (1.65 days with FCFS-F, 1.6 dayswith FCFS-S, and 1.4 days with EAHTS).

In Fig. 5, we observe that PFBM solutions outperform the FCFS andEAHTS policies as the congestion levels increase. Furthermore, Fig. 6shows that the ranges for the total service times of the schedulesproduced by PFBM are lower those resulted from FCFS policies. TheFCFS-F works better than the other three strategies under lowcongestion levels. As shown in both figures (via regression equations),there is a significant positive relationship between the performanceof the proposed policy and the congestion at the terminal.

5.1. MISH/MASH evaluation

In this subsection, we present a comparison of the MISH andMASH to CPLEX in solving the lower level problems. This comparisonis presented to showcase that both heuristic, developed to solve thelower level problems, provide optimal or near optimal solutions withsignificantly reduced CPU times. Thus, results of the simulation in

Table 1Dataset parameters.

Number ofberths

Inter-arrivaltime (h)

Arrival timewindowrange (h)

Handling timevolumeranges (h)

Handlingvolume set (%)

4 [3,4,5] [0,24] [8,24] [0.25,0.50,0.25][0,48] [24,40] [0.50,0.25,0.25]

[32,48] [0.25,0.25,0.50][0.33,0.33,0.33]

5 [3,4,5] [0,24] [8,24] [0.25,0.50,0.25][0,48] [24,40] [0.50,0.25,0.25]

[32,48] [0.25,0.25,0.50][0.33,0.33,0.33]

Table 2Average and range of cost per vessel (days) for 4 and 5 berths.

Average cost (days per vessel) Range (days per vessel)

PFBM FCFS-F FCFS-S EAHTS PFBM FCFS-F FCFS-S EAHTS4B/5B 4B/5B 4B/5B 4B/5B 4B/5B 4B/5B 4B/5B 4B/5B

0.79/0.58 0.61/0.46 0.68/0.53 1/0.73 0.12/0.18 0.71/1.17 0.74/1.13 0.76/1.151.02/0.73 0.94/0.4 1.04/0.46 1.18/0.84 0.15/0.15 0.76/0.75 0.78/0.74 0.8/0.821.1/0.89 0.97/0.65 1.11/0.73 1.34/1.11 0.13/0.21 1.05/1.38 1.04/1.35 1.08/1.231.43/0.91 1.3/0.78 1.44/0.86 1.67/1.19 0.2/0.19 1.06/1 1.04/0.99 1.02/1.051.45/1.14 1.23/0.86 1.38/0.95 1.72/1.32 0.18/0.2 1.19/0.86 1.17/0.85 1.03/0.841.46/1.17 1.39/0.96 1.54/1.11 1.61/1.41 0.17/0.17 1.1/1.07 1.08/1.02 0.99/1.051.46/1.21 1.69/0.98 1.83/1.09 1.71/1.39 0.17/0.24 1.03/1.09 1/1.09 0.88/0.951.49/1.22 1.3/0.97 1.44/1.08 1.91/1.45 0.24/0.12 1.21/0.58 1.17/0.57 1.22/0.591.5/1.24 1.47/1.13 1.58/1.28 1.7/1.51 0.18/0.25 0.98/1.06 0.99/1.05 0.94/1.131.56/1.25 1.61/1.16 1.73/1.33 1.84/1.71 0.16/0.32 0.96/1.18 0.92/1.16 0.85/1.081.65/1.32 1.79/1.2 1.89/1.38 1.76/1.69 0.19/0.21 1.07/1.53 1.05/1.45 1.02/1.381.67/1.52 1.71/1.61 1.84/1.82 1.96/1.95 0.18/0.22 1/1.5 0.97/1.42 1.03/1.541.79/1.56 1.86/1.3 2/1.44 1.94/1.72 0.24/0.19 1.18/0.88 1.17/0.89 1.02/0.821.85/1.59 1.87/1.31 2.01/1.44 2.1/1.68 0.21/0.16 1.14/0.68 1.12/0.68 1.04/0.781.85/1.63 1.85/1.66 1.99/1.79 2.2/1.91 0.22/0.17 1.11/1.06 1.09/1.01 1.25/1.031.98/1.7 1.9/1.49 2.05/1.63 2.21/2.02 0.27/0.2 1.17/0.7 1.15/0.7 0.99/0.732.18/1.85 2.26/1.79 2.41/1.95 2.47/2.32 0.22/0.28 1.23/1.2 1.18/1.11 1.03/1.152.23/2.02 2.27/2.17 2.45/2.32 2.57/2.41 0.34/0.15 1.41/0.58 1.39/0.63 1.27/0.782.24/2.1 2.44/2.14 2.63/2.34 2.61/2.53 0.24/0.25 1.36/1.12 1.33/1.06 1.33/1.042.26/2.33 2.18/2.16 2.32/2.45 2.59/2.66 0.3/0.18 1.26/0.93 1.23/0.86 1.23/0.832.34/2.41 2.32/2.56 2.5/2.83 2.76/2.77 0.27/0.33 1.43/1.37 1.45/1.23 1.34/1.252.4/2.49 2.46/2.79 2.64/2.99 2.74/3.02 0.24/0.21 1.19/0.92 1.16/0.93 1.03/0.962.45/2.53 2.72/2.55 2.86/2.81 2.77/2.9 0.27/0.14 1.36/0.97 1.34/0.91 1.34/0.892.82/2.92 3.35/3.14 3.59/3.4 3.18/3.52 0.28/0.24 1.65/0.94 1.6/0.9 1.4/0.87

M. Golias et al. / Computers & Operations Research 41 (2014) 412–422418

the previous section should not deviate if CPLEX was used instead ofthe MISH/MASH heuristics. The reason for not using CPLEX is thesavings in computational time. Table 3 provides the percentagedifference between the results obtained by the proposed heuristicsand the results obtained through CPLEX. As shown in Table 3, theresults for the objective function (13) using MISH compared to those

obtained via CPLEX are identical and MISH requires lower CPU times.In the case of the MASH, the percentage difference for the objectivefunctions range from 0% to 1%, that is, MASH provides solutionscloser to the solutions provided by CPLEX. The numerical resultsobtained with the MASH for the arrival window of one day providethe same solutions with CPLEX, while these results for a two days

Fig. 5. Average cost difference of PFBM to FCFS-F, FCFS-S, and EAHTS schedules.

M. Golias et al. / Computers & Operations Research 41 (2014) 412–422 419

arrival window deviate by 1% from the solutions of CPLEX. Further-more, the computational time for the MASH is significantly lowerthan the computational time required by CPLEX. For the 48 casesdescribed, the average computational time for MASH is less than0.002 s; whereas, the average computational time for the 48 casesusing CPLEX requires 1.37 s. Only three of the 48 cases had 1%

difference compared to the solutions of CPLEX. Nearly 94% of thetime (45 out of 48 cases), the results obtained were identical withCPLEX. Similarly, the computational time of the MISH (0.0006 s)is much smaller than the computational time required by CPLEX(1.39 s). Thus the MISH and MASH heuristics provide optimal or nearoptimal results with much less computational times.

Fig. 6. Average range difference of PFBM to FCFS-F, FCFS-S, and EAHTS schedules.

M. Golias et al. / Computers & Operations Research 41 (2014) 412–422420

6. Conclusions and future research

This paper presents a study that captures the realistic condi-tions of port operations by taking into account the inherentuncertainties of vessel arrivals and vessel handling times. Portoperators rely and depend on the development of accurate andeffective berth schedules, which should take into considerationmost of the factors that affect day to day operations. The industryis working towards keeping a track of metrics that will helpevaluation of the performance of port operations and carriersas well as the factors that may affect the smoothness of operations.Effective and efficient berth schedules, therefore, are essential forport operators and carriers. Modeling vessel arrival and handlingtimes as unknown variables brings the berth scheduling problemformulation closer to the reality because it can capture the uncer-tainty of weather conditions as well as the uncertainty of equip-ment breakdowns.

The problem is formulated as a bi-level bi-objective optimiza-tion problem to minimize the average and range of total servicetimes. A number of numerical results show that the proposedpolicy can provide, under uncertainty, more robust berth sche-dules when congestion effects become significant. As expected,when congestion levels are low, FCFS policies are efficient. Futureresearch is in progress to analyze the efficiency of the proposedapproach when probability distributions for vessel arrivals and/orhandling times are known and the applicability of the proposedframework to proactive/reactive scheduling approaches. Furtherfuture directions would include to study robustness of berthscheduling problems in case of continuous berth space.

Acknowledgment

This material is partially supported by the Intermodal FreightTransportation Institute (IFTI) and the Kathikas Institute of Researchand Technology (KIRT). Any opinions, findings, and conclusions orrecommendations expressed in this material are those of the authorsand do not necessarily reflect the views of IFTI or KIRT.

Appendix

IndicesiA I; I ¼ f1;…;ng Index for berths, there are n berths and I is

the set of berthsjA J; J ¼ f1;…;mg Index for vessels, there are m vessels and J is

the set of vesselsParameterscij Handling time of vessel j at berth icij

u Upper bound of cijcij

l Lower bound of cijC n�m matrix of cij valuesAj Arrival time of vessel jAj

u Upper bound of Aj

Ajl Lower bound of Aj

A m vector of Aj valuesDecisionvariables

xij 1 if vessel j is assigned to berth I, 0 otherwiseX n�m matrix of xij valuesyab 1 if xia ¼ xib ¼ 1 and b is the immediate

successor of a, 0 otherwiseY n�m matrix of yab valuesfj 1 if vessel j is the first vessel to be served at

its assigned berth, 0 otherwiselj 1 if vessel j is the last vessel to be served at

its assigned berth, 0 otherwisestj Service start time of vessel jftj Service finish time of vessel j

References

[1] Meisel F. Seaside operations planning in container terminals.Berlin/Heidel-berg: Physica-Verlag HD; 2009.

[2] Carlo HJ, Vis IFA, Roodbergen KJ. Seaside operations in container terminals:literature overview, trends, and research directions. Flexible Services andManufacturing Journal 2013. http://dx.doi.org/10.1007/s10696-013-9178-3.

Table 3CPLEX vs MISH and MASH for the lower level problem.

Dataset CPLEX vs MASH CPLEX vs MISH

Obj. fun. value difference CPU time CPLEX (s) CPU time heuristic (s) Obj. fun. value2 difference CPU time CPLEX (s) CPU time heuristic (s)

(1, 25) (0, 0) (1.16, 1.78) (0.019, 0.018) (0, 0) (1.39, 1.28) (0.0037, 0.0005)(2, 26) (0, 0) (1.45, 1.74) (0.002, 0.002) (0, 0) (1.24, 1.11) (0.0006, 0.0004)(3, 27) (0, 0) (1.42, 1.4) (0.001, 0.002) (0, 0) (1.09, 1.19) (0.0004, 0.0007)(4, 28) (0, 0) (1.12, 1.65) (0.001, 0.001) (0, 0) (1.31, 1.14) (0.0007, 0.0005)(5, 29) (0, 0) (1.48, 1.57) (0.001, 0.001) (0, 0) (1.1, 1.21) (0.0005, 0.0005)(6, 30) (0, 1%) (1.13, 1.4) (0.001, 0.002) (0, 0) (1.23, 1.12) (0.0004, 0.0005)(7, 31) (0, 0) (1.42, 1.56) (0.001, 0.002) (0, 0) (1.13, 1.33) (0.0005, 0.0004)(8, 32) (0, 0) (1.09, 1.51) (0.001, 0.002) (0, 0) (1.3, 1.23) (0.0004, 0.0004)(9, 33) (0, 0) (1.23, 1.39) (0.001, 0.002) (0, 0) (1.1, 1.17) (0.0004, 0.0004)(10, 34) (0, 0) (1.1, 1.56) (0.001, 0.002) (0, 0) (1.37, 1.29) (0.0004, 0.0004)(11, 35) (0, 0) (1.25, 1.61) (0.001, 0.001) (0, 0) (1.11, 1.12) (0.0004, 0.0004)(12, 36) (0, 1%) (1.14, 1.4) (0.001, 0.002) (0, 0) (1.4, 1.33) (0.0006, 0.0005)(13, 37) (0, 1%) (1.15, 1.48) (0.001, 0.001) (0, 0) (1.68, 1.41) (0.0005, 0.0005)(14, 38) (0, 0) (1.38, 1.45) (0.001, 0.002) (0, 0) (1.45, 1.68) (0.0005, 0.0005)(15, 39) (0, 0) (1.12, 1.43) (0.001, 0.001) (0, 0) (1.49, 1.67) (0.0005, 0.0005)(16, 40) (0, 0) (1.43, 1.48) (0.001, 0.001) (0, 0) (1.54, 1.48) (0.0004, 0.0006)(17, 41) (0, 0) (1.09, 1.43) (0.001, 0.001) (0, 0) (1.4, 1.48) (0.0004, 0.0006)(18, 42) (0, 0) (1.29, 1.39) (0.001, 0.001) (0, 0) (1.59, 1.47) (0.0006, 0.0004)(19, 43) (0, 0) (1.11, 1.53) (0.001, 0.001) (0, 0) (1.73, 1.45) (0.0004, 0.0004)(20, 44) (0, 0) (1.32, 1.49) (0.001, 0.001) (0, 0) (1.42, 1.66) (0.0007, 0.0005)(21, 45) (0, 0) (1.13, 1.42) (0.001, 0.001) (0, 0) (1.67, 1.41) (0.0004, 0.0005)(22, 46) (0, 0) (1.46, 1.54) (0.001, 0.001) (0, 0) (1.49, 1.87) (0.0006, 0.0005)(23, 47) (0, 0) (1.25, 1.47) (0.001, 0.001) (0, 0) (1.39, 1.77) (0.0004, 0.0004)(24, 48) (0, 0) (1.13, 1.38) (0.001, 0.001) (0, 0) (1.57, 1.6) (0.0005, 0.0006)

M. Golias et al. / Computers & Operations Research 41 (2014) 412–422 421

[3] Konur D, Golias MM. Cost-stable truck scheduling at a cross-dock facility withunknown truck arrivals. Transportation Research Part E: Logistics and Trans-port Review 2013;49:1.

[4] Golias MM. A bi-objective berth allocation formulation to account for vesselhandling time uncertainty. Journal of Maritime Economics and Logistics2011;13:419–41.

[5] Moorthy R, Teo C-P. Berth management in container terminal: the templatedesign problem. OR Spectrum 2006;28(4):495–518.

[6] Golias MM, Boile M, Theofanis S, The stochastic berth allocation problem. In:Proceedings of the second TRANSTEC conference prague, Czech Republic;2007.

[7] Zhou P., Kang H, Lin L, A dynamic berth allocation model based on stochasticconsideration. In: Proceedings of the 6th world conference on intelligentcontrol and automation. Dalian China: IEEE; 2006. p. 7297–301.

[8] Zhou P, Kang H. Study on berth and quay-crane allocation under stochasticenvironments in container terminal. Systems Engineering – Theory & Practice2008;28(1):161–9.

[9] Cartenì A, de Luca S. Simulation of a container terminal through a discreteevent approach: review and guidelines for application. In: Proceeding of theEuropean transport annual meeting: transportation planning methods; 2009.

[10] Kuo T, Huang W, Wu S, Cheng P. A case study of inter-arrival time distributionsof container ships. Journal of Marine Science and Technology 2006;14(3):155–64.

[11] Du Y, Xu Y, Chen Q, A feedback procedure for robust berth allocation withstochastic vessel delays. In: 2010 8th World congress on intelligent control andautomation (WCICA); 2010. p. 2210–5.

[12] Gao A, Zhang R, Du Y, Chen Q, A proactive and reactive framework for berthallocation with uncertainties. In: 2010 IEEE international conference onadvanced management science (ICAMS), vol. 3; 2010. p. 144–149.

[13] Xu Y, Chen Q, Quan X. Robust berth scheduling with uncertain vessel delayand handling time. Annals of Operations Research 2012;192:123–40.

[14] Zhen L, Lee LH, Chew EP. A decision model for berth allocation underuncertainty. European Journal of Operational Research 2011;212(Issue1):54–68 (ISSN 0377-2217, 10.1016/j.ejor.2011.01.021).

[15] Han XL, Lu ZQ, Xi LF. A proactive approach for simultaneous berth and quaycrane scheduling problemwith stochastic arrival and handling time. EuropeanJournal of Operational Research 2010:1327–40.

[16] Ben-Tal A, Nemirovski A. Robust convex optimization. Mathematics of Opera-tions Research 1998;23:769–805.

[17] Ngatchou P, Zarei A, El-Sharkawi MA, Pareto multiobjective optimization. In:2005 Proceedings of the 13th International Conference on Intelligent SystemsApplication to Power Systems; 2005. p. 84–91.

[18] Golias MM, The discrete and continuous berth allocation problem: models andalgorithms, dissertation. New Brunswick (NJ): Rutgers, The State University ofNew Jersey; 2007.

M. Golias et al. / Computers & Operations Research 41 (2014) 412–422422