Robust scheduling for Berth Allocation and Quay Crane Assignment Problem

Research Article Robust Scheduling for Berth Allocation and Quay Crane Assignment Problem M. Rodriguez-Molins, M. A. Salido, and F. Barber Instituto de Autom` atica e Inform` atica Industrial, Universitat Polit` ecnica de Val` encia,CaminodeVera,s/n,46022Val ` encia, Spain Correspondence should be addressed to M. A. Salido; [email protected].es Received 22 July 2014; Revised 24 November 2014; Accepted 1 December 2014; Published 31 December 2014 Academic Editor: Andrzej Swierniak Copyright © 2014 M. Rodriguez-Molins et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Decision makers must face the dynamism and uncertainty of real-world environments when they need to solve the scheduling problems. Different incidences or breakdowns, for example, initial data could change or some resources could become unavailable, may eventually cause the infeasibility of the obtained schedule. To overcome this issue, a robust model and a proactive approach are presented for scheduling problems without any previous knowledge about incidences. This paper is based on proportionally distributing operational buffers among the tasks. In this paper, we consider the berth allocation problem and the quay crane assignment problem as a representative example of scheduling problems. The dynamism and uncertainty are managed by assessing the robustness of the schedules. The robustness is introduced by means of operational buffer times to absorb those unknown incidences or breakdowns. Therefore, this problem becomes a multiobjective combinatorial optimization problem that aims to minimize the total service time, to maximize the buffer times, and to minimize the standard deviation of the buffer times. To this end, a mathematical model and a new hybrid multiobjective metaheuristic is presented and compared with two well-known multiobjective genetic algorithms: NSGAII and SPEA2+. 1. Introduction Within a container terminal, operations related to move containers can be divided into four different subsystems (shipto-shore, transfer, storage, and delivery/receipt) [1]. In each subsystem, terminal operators must deal with with different complex optimization problems that can be overcome by using artificial intelligence techniques. For instance, berthing allocation or stowage planning problems are related to the ship-to-shore area [2–5], remarshalling problem and transport optimization [6]tothestorageandtransfersubsystems, respectively, and planning and scheduling hinterland operations related to trains and trucks to the delivery/receipt subsystem [7]. In this paper, we focus on two problems related to the ship-to-shore area, the berth allocation problem (BAP) and the quay crane assignment problem (QCAP). The former is a well-known combinatorial optimization problem [8], which consists in assigning berthing positions and mooring times to incoming vessels. The QCAP deals with assigning a certain number of quay cranes (QCs) to each moored vessel such that all required movements of containers can be fulfilled [9]. A comprehensive survey of BAP and QCAP is given in [9]. These problems have been mostly considered separately, with an interest mainly focused on BAP. An interesting approach for BAP is presented by Kim and Moon [10]where a simulated annealing metaheuristic is compared with a mathematical model. However, there are some studies on the combined BAP + QCAP considering different characteristics of berths and cranes [11–15]. Most of the research in scheduling has been focused on deterministic and complete information, but they are usually notsatisfiedinreal-worldenvironments.Duetothefact that the real world is uncertain, imprecise, and nondeterministic, there might be unknown information, breakdowns, incidences, or changes, which make the initial plans or the obtained schedules become invalid. Thus, there are new trends to cope these aspects in the optimization techniques: proactive and reactive approaches [16]. In this paper, a proactive approach is studied within the berth allocation and Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 834927, 17 pages http://dx.doi.org/10.1155/2014/834927 2Mathematical Problems in Engineering the quay crane assignment problems. The uncertainty within these problems is due to lower movements per time unit than expected or engine failures in quay cranes, among others. Due to the introduction of this new objective in the scheduling optimization problem, a multiobjective optimization approach needs to be taken into consideration. All the above studies do not take into consideration the uncertainty of the real world to obtain a robust scheduling. Robustness is a measure of the performance characterization of an algorithm in the presence of uncertainties [17]. However, there are some studies that address the robust scheduling. In [18], a robust optimization model for cyclic berthing for a continuous and dynamic BAP is studied by minimizing the maximal crane capacity over different arrival scenarios of a bounded uncertainty given by their arrival agreements. In [19], a proactive approach for a discrete and dynamic model of the BAP is presented taking into account uncertainties in the arrival and handling times given their probability density functions. They propose a mixed integer programming model and a genetic algorithm (GA) for both problems: discrete berth allocation and QC assignment. The objective is to minimize the sum of expected value, the standard deviation of the service time, and the tardiness of the incoming vessels. Robust scheduling based on operational buffers has already been introduced as a proactive approach in the BAP. An approach to robust BAP is presented in [20]. They presented a feedback procedure for the BAP that iteratively improves the robustness of the initial schedule. This feedback procedure determines the time buffers for each vessel by means of adjustment rules. In [21], another approach to the robust BAP is solved by a scheduling algorithm that integrates simulated annealing and branch-and-bound algorithms. This study introduces the robustness as an objective to be maximized and an evaluation is carried out by varying the weights of these functions. The robustness is achieved by a constant buffer time assigned to all vessels. In [22], the robust BAP problem is studied as a proactive strategy as a multiobjective optimization problem. They solved this problem with the squeaky wheel optimization (SWO) metaheuristic. The first objective is to minimize the late departures and the deviation from the desired position; the second objective is to maximize the robustness of the schedule. They tackle the robustness measure as a diminishing return, specifically the exponential function, to capture the decreasing marginal productivity of slacks in a berthing schedule. However, most of the above approaches consider discrete berths or previous knowledge about the uncertainty in arrival or handling times to produce robust schedules, but usually this knowledge is not available. Furthermore, other approaches propose how to obtain robust schedules by means of operational buffer times, but these buffers are set independently of the handling (or processing) time of the vessels. Overcoming the above approaches, hybrid metaheuristics for both single and multiobjective combinatorial optimization problems have received a significant interest from the research community [23,24],andalsotheyhavebeenused in a wide range of real-world applications [25]. In this paper, we introduce a robust model to deal with limited incidences with no previous knowledge about them (Section 3)aswellasamultiobjectiveapproachtofacethis problem (Section 4). A formal mixed integer programming (MIP)ispresentedforthedynamicandcontinuousrobust BAP + QCAP that extends the model presented in [10] (Section 5). Section 6presents our proposed hybrid multiobjective genetic algorithm based on the scheme NSGAII [26] in order to obtain near-optimal solutions in an efficient way. This hybrid algorithm is used to solve the BAP + QCAP with a continuous quay and dynamic arrivals as well as to provide robust solutions by using operational buffers. As there is no previous knowledge about the incidences, these operational buffers are proportionally distributed among the tasks to be able to absorb as many incidences as possible. Thereby, a new objective function (standard deviation of robustness measures) was introduced to pursue this goal. This algorithm is compared with the mathematical model presented and two well-known multiobjective genetic algorithms: NSGAII and SPEA2+ [27](Section7). The development of the technique presented in this paper will provide the terminal operators with different robust berthing plans which are able to absorb limited incidences. The overall collaboration goal of our group at the Universitat Polit` ecnica de Val` encia (UPV) with the Valencia Port Foundation and the maritime container terminal MSC (Mediterranean Shipping Company S.A.) is to offer assistance and help in the planning and scheduling of tasks such as the allocation of spaces to outbound containers, to identify bottlenecks, to determine the consequences of changes, to provide support in the resolution of incidents, and to provide alternative berthing plans. Thus, the development of the technique presented in this paper will provide the terminal operators with different robust berthing plans which are able to absorb limited incidences. 2. Berthing Allocation and Quay Crane Assignment: BAP + QCAP Let 𝑉be a set of incoming vessels; BAP + QCAP consists in obtaining an optimal (or near-optimal) schedule of the vessels 𝑉by assigning mooring times, berthing positions, and QCs to each vessel. Our BAP + QCAP model is classified, according to the classification given by Bierwirth and Meisel [9] as follows. (i) Spatial Attribute: Continuous Layout. We assume that the quay is a continuous line, so there is no partitioning of the quay and the vessel can berth at arbitrary positions within the boundaries of the quay. It must be taken into account that, for a continuous layout, berth planning is more complicated than for a discrete layout, but it better utilizes the quay space [9]. (ii) Temporal Attribute: Dynamic Arrival. Fixed arrival times are given for the vessels, so that vessels cannot berth before their expected arrival times. Mathematical Problems in Engineering 3 Quay (m) L 600 500 400 300 200 100 2468101214 aimi 𝓁ili wi 𝜂i 𝜂i pi di hi i Time (hr) Figure 1: Data related to one vessel. (iii) Handling Time Attribute: Unknown in Advance.The handling time of a vessel depends on the number of assigned QCs (QCAP)andthemovesrequired. (iv) Performance Measure: Wait and Handling Times.The objective is to minimize the sum of the waiting and handling times of all vessels 𝑉. Following, we introduce the notation used for each vessel 𝑖∈𝑉(Figure 1). The integer data variables are (i) 𝐾: total number of QCs in the container terminal. We assume all QCs carry out the same number of movements per time unit (movsQC),givenbythe container terminal, (ii) 𝐿: total length of the berth in the container terminal, (iii) 𝑎𝑖: arrival time of the vessel 𝑖at port, (iv) 𝑐𝑖:numberofrequiredmovementstoloadandunload containers of vessel 𝑖, (v) ℓ𝑖: vessel length. The decision variables are (i) 𝑚𝑖:mooringtimeof𝑖.Thus,waitingtime(𝑤𝑖)ofvessel 𝑖is calculated as (𝑤𝑖=𝑚𝑖−𝑎𝑖), (ii) 𝑝𝑖: berthing position where vessel 𝑖moors, (iii) 𝑞𝑖: number of assigned QCs to vessel 𝑖, (iv) 𝑢𝑖𝑘: indicates whether the QC 𝑘(1≤𝑘≤𝐾)works (1)or not (0) on the vessel 𝑖, (v) 𝑛𝑖𝑘: denotes that the number of QCs assigned to vessel 𝑖is 𝑘QCs (𝑛𝑖𝑘 =1), For instance, if vessel 3 has been assigned 4 QCs, then 𝑛34 =1and the others QCs 𝑛3𝑘 =0,∀𝑘=1,...,𝐾,𝑘 =4. The variables derived from the previous ones are (i) 𝐻𝑖𝑘: loading and unloading time at quay (handling time) of vessel 𝑖using 𝑘QCs (1≤𝑘≤𝐾). This handling time depends on 𝑐𝑖and it is defined by 𝐻𝑖𝑘 =⌈ 𝑐𝑖 𝑘∗movsQC ⌉ ∀𝑖∈𝑉,∀𝑘=1,...,𝐾, (1) (ii) ℎ𝑖: required handling time of vessel 𝑖when 𝑞𝑖QCs are assignedtoit.Thisvalueissetbymeansof𝐻𝑖𝑞𝑖, (iii) 𝑡𝑖𝑘: working time of the 𝑘th QC (1≤𝑘≤𝐾)thatis assigned to vessel 𝑖, (iv) 𝑑𝑖: departure time of vessel 𝑖(𝑑𝑖=𝑚𝑖+ℎ𝑖), (v) 𝑠𝑖and 𝑒𝑖: indexes of the first and last QC assigned to vessel 𝑖,respectively. In this study, the following assumptions are considered. (i) All the information related to the waiting vessels is known in advance (arrival, priority, moves, and length). (ii) Every vessel has a draft that is lower than or equal to the draft of the quay. (iii) Movements of QCs along the quay as well as berthing and departure times of vessels are not considered since it supposes a constant penalty time for all vessels. (iv) Simultaneous berthing is allowed, subject to the length of the berth. Usually in container terminals, the number of QCs could vary during execution at the quay. This issue has been studied in Rodriguez-Molins et al. [5]. However, in this paper and without loss of generality, we study the robustness of the schedules assuming that the number of QCs assigned to one vessel does not vary along the moored time. Once a QC starts ataskinavessel,itmustcompleteitwithoutanypauseor shift (nonpreemptive tasks). Thus, all QCs assigned to the same vessel 𝑖havethesameworkingtimeonthevessel(𝑡𝑖𝑘 = ℎ𝑖,∀𝑘=1,...,𝐾,𝑢𝑖𝑘 =1). The following constraints must be accomplished. (i) Moored time of vessel 𝑖must be at least the same that its arrival time (𝑚𝑖≥𝑎𝑖). (ii) There is a safe distance between two moored ships. We assume that each vessel 𝑖has a 2.5% of this length at each side as a safe distance (𝜂𝑖)(Figure 1). This safe distance is added to the length of each vessel 𝑖:𝑙𝑖:= ℓ𝑖+2𝜂𝑖. (iii) There must be enough contiguous space at berth to moor a vessel 𝑖of length (𝑙𝑖). (iv) There must be at least one QC assigned to each vessel. Furthermore, there is a maximum number of QCs that can be assigned to vessel 𝑖(QC+ 𝑖). This value, (QC+ 𝑖), depends on the length of each vessel (ℓ𝑖), since a safe distance is required between two contiguous 4Mathematical Problems in Engineering QCs (safeQC)and the maximum number of QCs that the container terminal allows per vessel (maxQC) (equtaion (2)). Both safeQC and maxQC parameters aregivenbythecontainerterminal: QC+ 𝑖=min (maxQC,max (1,⌊ ℓ𝑖 safeQC ⌋)) ∀𝑖∈𝑉. (2) Our objective is to allocate all vessels according to several constraints minimizing the total waiting (𝑇𝑤) and handling or processing time (𝑇ℎ), known as the service time (𝑇𝑠), for all vessels: 𝑇𝑤=∑ 𝑖∈𝑉𝑤𝑖,(3) 𝑇ℎ=∑ 𝑖∈𝑉ℎ𝑖,(4) 𝑇𝑠=𝑇𝑤+𝑇𝑠.(5) 3. Robust BAP + QCAP Model Uncertainty and nondeterminism of real-world environments may cause difficulties in the initial plans made by the decision makers. In container terminals, the initial obtained schedules for the BAP + QCAP problem might become invalid due to different reasons: breakdowns in QCs, late arrivals of the vessels, extreme weather events, a lower ratio of movements per QC than expected, and so forth. The robustness concept means that, given a schedule, this initial schedule remains feasible when minor incidences occur in its actual scenario. The usual disruptions to be considered in BAP + QCAP are the following: (i) early or late arrival of a vessel 𝑖from its expected arrival time (𝑎𝑖); (ii) the handling time of a vessel 𝑖is larger than its expected handling time (ℎ𝑖). In this paper, we focus just on the disruptions affecting the handling time which eventually delay the departure time. In case of incidences related to late arrivals, they could also be modeled as delays in the handling time of the vessels which eventually also delay their departure time. Definition 1. Given the possible disruptions, we consider that ascheduleisrobustifadisruptioninonevesseldoesnotaffect or alter the mooring times of the other vessels. The robustness of a schedule of BAP + QCAP might be guaranteed through two periods of time related to each vessel: waiting time of a vessel (𝑤𝑖)andbuffertimeafterthe departure of each vessel (𝑏𝑖)[28]. Without loss of generality, early arrivals are not taken into account since they only increase waiting times but they do not alter mooring times. Theschedulecouldabsorbdelaysorbreakdownsthatdo not exceed the sum of those two periods (𝑤𝑖+𝑏𝑖). Therefore, both times should be maximized in order to achieve the Quay (m) L 2 4 3 1 h1 h2b2 b1 h3 h4 b3 b4 Time (hr) 024681012141618202224 Figure 2: Buffer times 𝑏𝑖givenanexampleschedule. maximum robustness and ensure that there is no need to rescheduletheinvolvedvessels.However,itshouldbekept in mind that the first objective of the BAP + QCAP is to minimize the total service time of the incoming vessels (𝑤𝑖+ ℎ𝑖). Therefore, following the proposal given by Davenport et al. [29], we focus on maximizing only the second period of time, buffer times (∑𝑏𝑖),toobtainrobustschedules. Let 𝜑𝑖be the vessels that succeed vessel 𝑖and occupy some berth space of vessel 𝑖(𝜑𝑝 𝑖)oruseanyofQCsassignedto vessel 𝑖(𝜑𝑞 𝑖). The buffer time of vessel 𝑖(𝑏𝑖)istheminimum difference (𝜏𝑖𝑗) between the departure time of vessel 𝑖(𝑑𝑖)and the mooring time of vessel 𝑗(𝑚𝑗,𝑗∈𝜑𝑖). In case there is no vessel in 𝜑𝑖, the maximum buffer time is assigned to 𝑏𝑖(an infinite value). Figure 2shows an example of the buffer times (𝑏𝑖)assigned for each scheduled vessel as an empty rectangle: 𝜑𝑝 𝑖={∀𝑗∈𝑉,𝑚𝑗≥𝑑𝑖 ∧[𝑝𝑖,𝑝𝑖+𝑙𝑖)∩[𝑝𝑗,𝑝𝑗+𝑙𝑗) =⌀} ∀𝑖∈𝑉 𝜑𝑞 𝑖={∀𝑗∈𝑉,𝑚𝑗≥𝑑𝑖∧∃𝑘,1≤𝑘≤𝐾 ∧𝑢𝑖𝑘 =1∧𝑢𝑗𝑘 =1} ∀𝑖∈𝑉 𝜑𝑖=𝜑𝑝 𝑖∪𝜑𝑞 𝑖∀𝑖∈𝑉 𝜏𝑖𝑗 =𝑚𝑗−𝑑𝑖∀𝑖∈𝑉,∀𝑗∈𝜑𝑖 𝑏𝑖={ { {+∞, 󵄨󵄨󵄨󵄨𝜑𝑖󵄨󵄨󵄨󵄨=0 min 𝑗∈𝜑𝑖(𝜏𝑖𝑗), otherwise ∀𝑖∈𝑉. (6) In this paper, we assume that the more handling time is, the more likely it is to suffer incidences. Therefore, in general, the larger the buffers are, the more robust the schedules are. Nevertheless, regarding the concept of decreasing productivity (or diminishing returns) presented in [22], there is a certain buffer size beyond which no more robustness is added to the schedule. Thereby, there is no need to assign large buffer times to each vessel. For instance, in Figure 2,vessel 1 would not need 8 time units of buffer time (𝑏1)since its handling time is only 3 time units. It is not likely that this vessel would suffer a delay of that magnitude. However, vessel 2, with a handling time of 8 time units, has only 2 time units of buffer time (𝑏2). In this case, it is highly likely that this vessel Mathematical Problems in Engineering 5 3 (4) 1 (4) 7 (5) 9 (5) 5 (5) 4 (2) 25 25 24 37 39 21 52 30 6 (5) 8 (2) 2 (3) 700 600 500 400 300 200 100 50 100 150 200 250 300 (a) Robust schedule 3 1 (5) 5 5 6 (5) 9 (5) 5 (5) 8 (5) 17 6 (5) 4 (4) 7 (2) 2 (4) 700 600 500 400 300 200 100 50 100 150 200 250 300 (b) Optimal schedule according to 𝑇𝑠 Figure 3: Two possible schedules given the same incoming vessels. suffers some breakdown or delay and so it becomes invalid this schedule. Furthermore, we consider that the magnitude of the incidence is related to the handling time of the vessel. Thus, the robustness measure of each vessel 𝑖(𝑟𝑖∈[0,1])isrelated to the buffer time 𝑏𝑖and the average handling time ℎ∗ 𝑖(equation (7)). It should be mentioned that other functions, for example, exponential function, could be adopted to define the robustness of each vessel: ℎ∗ 𝑖=𝑐𝑖 ((1+QC+ 𝑖)/2)movsQC .(7) Given the robustness of each vessel, the robustness of a schedule 𝑅∈[0,|𝑉|]is defined by (9),where𝜔𝑖is a weighting factor (𝜔𝑖≥1) which depends on historical data, if available. A𝜔𝑖=1value represents that vessel 𝑖used to finish its tasks as expected, and 𝜔𝑖>1value denotes that vessel 𝑖used to be delayed: 𝑟𝑖=min (1, 𝑏𝑖 𝜔𝑖ℎ∗ 𝑖), ∀𝑖∈𝑉, (8) 𝑅=∑ 𝑖∈𝑉𝑟𝑖.(9) In this paper, we address the BAP + QCAP problem without knowledge of the incidences; thus, the weighting factor is the same for all the vessels (𝜔𝑖=1,∀𝑖∈𝑉). Example 2. Figure 3shows two different schedules given the same set of 9 incoming vessels. Each vessel is labeled with its vessel’s ID and the assigned QC number in brackets. Furthermore, the buffer time between vessels is also showed. On the one hand, Figure 3(a) represents a robust schedule since limited incidences over any vessel could be absorbed. On the other hand, Figure 3(b) shows a schedule with the optimal solution according to the objective function 𝑇𝑠.The latter schedule will be highly likely unfeasible if any incidence occurs. Figures 3(a) and 3(b) are an example of the well-known trade-off between optimality and robustness. However, a robust schedule is not only achieved by extending an optimal schedule over the time. A robust schedule must also consider an optimized allocation of vessels to achieve the maximum sum of buffer sizes with a proper distribution among all vessels. Note that the optimality is not directly the makespan oftheschedulebutthetotalservicetime(waitingand handling times). An important issue in this paper is that there is no available information about how likely the incidences or breakdowns occur. Therefore, it is interesting that these buffers are proportionally distributed among all the vessels. Thereby, a third objective is introduced into the model in order to improve the robustness of a schedule: minimizing the standard deviation (𝜎) of the robustness measures of all vessels (𝑟𝑖∀𝑖∈𝑉): 𝜎=√1 |𝑉|∑ 𝑖∈𝑉(𝑟𝑖−𝑟), (10) where 𝑟is the average of the buffers of the schedule and |𝑉|is the number of incoming vessels. Both measures presented above, robustness of a schedule (𝑅) and standard deviation of these values (𝜎), represent 6Mathematical Problems in Engineering 5 1 (3) 36 328 (5) 9 (4) 8 (4) 10 (5) 4 (5) 93 20 50 6 (5) 3 (4) 7 (3) 2 (2) 700 600 500 400 300 200 100 50 100 150 200 250 300 350 4 00 (a) High standard deviation 5 1 (3) 25 28 19 (5) 9 (4) 8 (4) 6 (4) 4 (5) 70 13 42 30 30 3 (4) 7 (3) 2 (2) 10 (3) 700 600 500 400 300 200 100 50 100 150 200 250 300 350 4 00 (b) Low standard deviation Figure 4: Two different schedules with similar robustness and different standard deviation. the actual robustness of a schedule Rto be maximized (see (11)). This measure guarantees the absorption of incidences that imply at most a delay of a R%oftheweightedaverage handling time (𝜔𝑖ℎ∗ 𝑖): R=𝑅−𝜎. (11) Example 3. Figure 4shows two different schedules with a similar robustness value (𝑅 = 0.7) but different standard deviations, 𝜎1=0.17and 𝜎2=0.45. With these values, the firstschedulehasanactualrobustnessvalueofR1=0.7− 0.17=0.53.Thus,inaverage,thisscheduleguaranteesthat it could absorb incidences that imply at most a delay of the 53% of the average handling time of the vessels. In contrast, thesecondschedulehasanactualrobustnessvalueofR2= 0.7−0.45=0.25; thus, it is able to absorb only incidences that imply at most a 25% of the average handling time of the vessels. Surico et al. [30] presented a close function to measure the robustness of a schedule (avg(𝑤𝑖)−𝛼𝜎(𝑤𝑖),∀𝑖∈𝑉). avg(𝑤𝑖) and 𝜎(𝑤𝑖)denote the average and the standard deviation of the waiting times, respectively; 𝛼is a constant weighting factor that must be set. However, this measure does not reflect the relationship between the handling or processing time of the task and the buffer times. Thus, to our best knowledge, there is no other study which, considering the BAP+QCAPwithacontinuousquayanddynamicarrivals, tackles the robustness without any previous knowledge about the incidences. Figure 4showstwodifferentschedulesof10vesselswith the same high value of the robustness measure. However, schedule of Figure 4(a) has a greater value for the standard deviation (𝜎)thanscheduleofFigure4(b).Thereby,itis importanttonotethatbuffertimesfromscheduleinFigure 4(a) are not equally distributed and this schedule will fail if an incidence which delays the departure time just 1 time unit over vessels 4 or 6 occurs or more than 3 time units over vessel 1. However, in the schedule in Figure 4(b),itishighly unlikely to be invalid since all vessels have enough buffer time after its schedule departure time. 4. Multiobjective Approach for the Robust BAP + QCAP Three different objectives must be optimized to solve the robust BAP + QCAP: the service time (𝑇𝑠)(equation (5)), the robustness (𝑅)(equation (9)), and the standard deviation of the robustness measures 𝜎(𝑅)(equation (10)). These objective functions must be normalized in order to apply the search process correctly. Equation (14) shows how to normalize the service time objective into the interval [0,1]( 𝑇𝑠) and it implies to normalize both the waiting time  𝑇𝑤(equation (12))andthehandling time  𝑇𝑠(equation (13)). On the one hand, the handling time is just a linear normalization since the maximum (ℎ+ 𝑖)and minimum (ℎ− 𝑖)times are known by assigning the minimum (1)andthemaximumnumberofQCstovessel𝑖(QC+ 𝑖). On the other hand, normalizing the waiting time requires to determine a maximum total waiting time (𝑊𝐹). In this case, 𝑊𝐹value is the total waiting time of the incoming vessels when a first-come, first-served (FCFS) policy is applied, assigning 2 QCs to each vessel, and just one vessel is allowed in the berth at the same time (see, for example, Figure 5). The maximum total waiting time (𝑊𝐹)couldalsobeobtainedby assigning just one QC to each incoming vessel, but in that case, 𝑊𝐹value would be too large and all the normalized waiting times would be close to zero:  𝑇𝑤=1 𝑊𝐹∑ 𝑖∈𝑉(𝑚𝑖−𝑎𝑖) 𝑇𝑤∈[0,1],(12) Mathematical Problems in Engineering 7 1 (2) (2)(2) (2)(2) 23 45 50 100 150 200 250 300 350 400 450 500 550 700 600 500 400 300 200 100 Figure 5: Schedule generated to obtain the maximum value of waiting time 𝑊𝐹.  𝑇ℎ=1 |𝑉|∑ 𝑖∈𝑉(ℎ𝑖−ℎ− 𝑖 ℎ+ 𝑖−ℎ− 𝑖) 𝑇ℎ∈[0,1],(13)  𝑇𝑠= 𝑇𝑤+ 𝑇ℎ 2 𝑇𝑠∈[0,1].(14) Robustness objective function must also be normalized into the interval [0,1]( 𝑅) as defined by (15).Thethirdobjective, standard deviation of robustness measures, is already normalized due to the fact that 𝑟𝑖values are already in the interval [0,1](equation (10)):  𝑅=𝑅 |𝑉| 𝑅∈[0,1].(15) Thereby, the objective function for the robust BAP + QCAP is to minimize the function 𝐹(equation (16)). Each coefficient 𝜆𝑖(0 ≤ 𝜆𝑖≤1)assigns different weights to each component or objective function in order to establish an aggregate function: 𝐹=𝜆1 𝑇𝑠−𝜆2 𝑅+𝜆3𝜎. (16) These coefficients 𝜆𝑖are subject to ∑𝑖𝜆𝑖=1. In a multiobjective optimization problem, usually there is no single solution wherein all its objectives are simultaneously optimized. However, there may exist a set of Pareto optimal solutions with different trade-offs between their objective functions. Pareto efficiency, or Pareto optimality, is a solution in which it is impossible to make any one criterion better off without making at least one criterion worse off [31]. Pareto optimal solutions are defined by means of the dominance concept. Considering the robust BAP + QCAP, let 𝑥and 𝑦be two different solutions; 𝑥dominates 𝑦if at least one of the following conditions is satisfied:  𝑇𝑠(𝑥)< 𝑇𝑠(𝑦) ∧ 𝑅(𝑥)≥ 𝑅(𝑦) ∧𝜎(𝑥)≤𝜎(𝑦),  𝑇𝑠(𝑥)≤ 𝑇𝑠(𝑦) ∧ 𝑅(𝑥)> 𝑅(𝑦) ∧𝜎(𝑥)≤𝜎(𝑦),  𝑇𝑠(𝑥)≤ 𝑇𝑠(𝑦) ∧ 𝑅(𝑥)≥ 𝑅(𝑦) ∧𝜎(𝑥)<𝜎(𝑦). (17) Given a set of feasible solutions 𝐷,asolution𝑥∈𝐷is Paretooptimalsolutionifitisnondominatedbyanyother solution 𝑥󸀠∈𝐷.TheParetooptimalsetisthesetofallthe solutions that are Pareto optimal solutions [31]. In general, generating the Pareto optimal set is expensive computationally and it is often impracticable. Therefore, algorithms try to find a good approximation of the Pareto optimalset.Inthiswork,wereferthateachapproximation as Pareto front, which contains solutions that, although are nondominated among them, could be dominated by other solutions not found by our algorithms. 5. Mathematical Formulation A mixed integer programming (MIP) model is presented to solvetherobustBAP+QCAP.Theobjectivefunctionofthis model is to minimize (16). This mathematical model is based onthemodelpresentedin[10,28]. In the proposed model, 𝑀denotes a sufficiently large number (as it is used in MIP). Furthermore, there are four auxiliary binary variables. 𝑧𝑥 𝑖𝑗 is a decision variable that indicates if vessel 𝑖is located to the left of vessel 𝑗on the berth (𝑧𝑥 𝑖𝑗 =1); 𝑧𝑦 𝑖𝑗 =1indicates that vessel 𝑖is moored before vessel 𝑗in time. The auxiliary variable 𝑢𝑖𝑘 indicates whether the QC 𝑘works (1)or not (0) on vessel 𝑖;𝑛𝑖𝑘 =1denotes that the number of QCs assigned to vessel 𝑖is 𝑘: ∀𝑖,𝑗∈𝑉 𝑖 =𝑗 ∀𝑘=1,...,𝐾 𝑧𝑥 𝑖𝑗,𝑧𝑦 𝑖𝑗,𝑢𝑖𝑘,𝑛𝑖𝑘0/1integer.(18) In the proposed model, there are four auxiliary binary variables. 𝑧𝑥 𝑖𝑗 is a decision variable that indicates if vessel 𝑖is located to the left of vessel 𝑗on the berth (𝑧𝑥 𝑖𝑗 =1); 𝑧𝑦 𝑖𝑗 =1 indicates that vessel 𝑖is moored before vessel 𝑗in time. The auxiliary variable 𝑢𝑖𝑘 indicates whether the QC 𝑘works (1)or not (0) on vessel 𝑖;𝑛𝑖𝑘 =1denotes that the number of QCs assigned to vessel 𝑖is 𝑘. The constraints of this mathematical model are detailed below. Constraint (19) ensures that vessels must moor afer they arrive at the terminal: ∀𝑖∈𝑉 𝑚𝑖≥𝑎𝑖.(19) Constraints (20) and (21) establish the waiting and departure times according to 𝑚𝑖and ℎ𝑖: ∀𝑖∈𝑉 𝑤𝑖=𝑚𝑖−𝑎𝑖,(20) ∀𝑖∈𝑉 𝑑𝑖=𝑚𝑖+ℎ𝑖.(21) Constraint (22) guarantees that a moored vessel does not exceed the length quay: ∀𝑖∈𝑉 𝑝𝑖+𝑙𝑖≤𝐿. (22) The number of QCs to the vessel 𝑖are assigned by means of constraints (23)–(28) as follows: ∀𝑖∈𝑉 𝑞𝑖=𝐾 ∑ 𝑘=1𝑢𝑖𝑘,(23) ∀𝑖∈𝑉 𝐾 ∑ 𝑘=1𝑛𝑖𝑘 =1, (24) 8Mathematical Problems in Engineering ∀𝑖∈𝑉 𝐾 ∑ 𝑘=1𝑛𝑖𝑘𝑘=𝑞𝑖,(25) ∀𝑖∈𝑉 1≤𝑞𝑖≤QC+ 𝑖,(26) ∀𝑖∈𝑉 1≤𝑠𝑖≤𝑒𝑖≤𝐾, (27) ∀𝑖∈𝑉 𝑞𝑖=𝑒𝑖−𝑠𝑖+1. (28) Constraints (29)–(31) establish the minimum handling time needed to load and unload their containers according to the number of assigned QCs: ∀𝑖∈𝑉 𝐾 ∑ 𝑘=1𝑡𝑖𝑘 movsQC ≥𝑐𝑖(29) ∀𝑖∈𝑉 𝐾 ∑ 𝑘=1𝑛𝑖𝑘𝐻𝑖𝑘 =ℎ𝑖(30) ∀𝑖∈𝑉 ℎ𝑖=max ∀𝑘=1,...,𝐾𝑡𝑖𝑘.(31) Constraint (32) ensures that QCs that are not assigned to vessel 𝑖have 𝑡𝑖𝑘 =0: ∀𝑖∈𝑉∀𝑘=1,...,𝐾 𝑡𝑖𝑘 −𝑀𝑢𝑖𝑘 ≤0. (32) Constraint (33) forces all assigned QCs to vessel 𝑖working the same number of hours: ∀𝑖∈𝑉∀𝑘=1,...,𝐾 ℎ𝑖−𝑀(1−𝑢𝑖𝑘)−𝑡𝑖𝑘 ≤0. (33) Constraint (34) avoids that one QC is assigned to two different vessels at the same time: ∀𝑖,𝑗∈𝑉∀𝑘=1,...,𝐾 𝑢𝑖𝑘 +𝑢𝑗𝑘 +𝑧𝑥 𝑖𝑗 ≤2. (34) Constraints (35) and (36) forcetheQCstobecontiguously assigned (from 𝑠𝑖up to 𝑒𝑖): ∀𝑖∈𝑉∀𝑘=1,...,𝐾 𝑀(1−𝑢𝑖𝑘)+(𝑒𝑖−𝑘)≥0, (35) ∀𝑖∈𝑉∀𝑘=1,...,𝐾 𝑀(1−𝑢𝑖𝑘)+(𝑘−𝑠𝑖)≥0. (36) The safety distance between vessels is taken into account by constraint (37) as follows: ∀𝑖,𝑗∈𝑉 𝑖 =𝑗 𝑝𝑖+𝑙𝑖≤𝑝𝑗+𝑀(1−𝑧𝑥 𝑖𝑗). (37) Constraint (38) avoids that one vessel uses a QC which should cross through the others QCs: ∀𝑖,𝑗∈𝑉 𝑖 =𝑗 𝑒𝑖+1≤𝑠𝑗+𝑀(1−𝑧𝑥 𝑖𝑗). (38) Constraint (39) avoids that vessel 𝑗moors while the previous vessel 𝑖is still at the quay: ∀𝑖,𝑗∈𝑉 𝑖 =𝑗 𝑑𝑖≤𝑚𝑗+𝑀(1−𝑧𝑦 𝑖𝑗). (39) Constraint(40) establishes the relationship between each pair of vessels avoiding overlaps: ∀𝑖,𝑗∈𝑉 𝑖 =𝑗 𝑧𝑥 𝑖𝑗 +𝑧𝑥 𝑗𝑖 +𝑧𝑦 𝑖𝑗 +𝑧𝑦 𝑗𝑖 ≥1. (40) Constraint (41) ensures that the total waiting time of the schedule does not exceed the maximum total waiting time (𝑊𝐹):∑ 𝑖∈𝑉𝑤𝑖≤𝑊𝐹.(41) Constraints (42)–(44) assign the time between the departure time of vessel 𝑖and the mooring time of vessel 𝑗.Forthose vessels 𝑗so that 𝑧𝑡 𝑖𝑗 =1,theyareassigned𝑀as a value representing an unbounded time for the robustness: ∀𝑖,𝑗∈𝑉 𝑖 =𝑗 𝑧𝑡 𝑖𝑗 =𝑧𝑥 𝑖𝑗 +𝑧𝑥 𝑗𝑖 +𝑧𝑦 𝑖𝑗,(42) ∀𝑖,𝑗∈𝑉 𝑖 =𝑗∧(𝑧𝑡 𝑖𝑗=0∨𝑧𝑡 𝑖𝑗=2) 𝜏𝑖𝑗 =𝑀, (43) ∀𝑖,𝑗∈𝑉 𝑖 =𝑗∧𝑧𝑡 𝑖𝑗=1 𝑑𝑖+𝜏𝑖𝑗 =𝑚𝑗+𝑀(1−𝑧𝑦 𝑖𝑗). (44) Constraints (45) and (46) set the value of the available buffer time after vessel 𝑖and its robustness value, respectively: ∀𝑖∈𝑉 𝑏𝑖=min (min 𝑗∈𝑉 𝑖 =𝑗 (𝜏𝑖𝑗),ℎ∗ 𝑖),(45) ∀𝑖∈𝑉 𝑟𝑖ℎ∗ 𝑖=𝑏𝑖.(46) The decision variable 𝑧𝑡 𝑖𝑗 (see constraint (47)) indicates if a vessel 𝑗moors later than 𝑖and, at the same time, the vessel 𝑗 intersects with the berth length occupied by vessel 𝑖(𝑧𝑡 𝑖𝑗): ∀𝑖,𝑗∈𝑉 𝑖 =𝑗 0≤𝑧𝑡 𝑖𝑗 ≤2 (47) 6. Multiobjective Genetic Algorithms: MOGA + SA Commonly approximations of the Pareto optimal sets of a multiobjective optimization problem are obtained by means of multiobjective evolutionary algorithms [31]. Furthermore, nowadays, metaheuristics are usually hybridized with other techniques or algorithms in order to enhance their effectiveness and performance [23,24]. One of the most common forms of hybrid genetic algorithm involves incorporating local search to a canonical genetic algorithm. Genetic algorithm is used to perform global exploration among the population, and local search is used to perform local exploitation around the chromosomes. Because of the complementary properties of genetic algorithms and local search methods, the hybrid approach often outperforms either methods operating alone [32]. Thereby, a hybrid multiobjective genetic algorithm (MOGA) has been implemented in this paper. The NSGAII Mathematical Problems in Engineering 9 Vessel identifier (i) Number of cranes (qi) Buffer size (bi) Figure 6: Structure of one gene of a chromosome. schema has been extended with a multiobjective local search based on the multiobjective simulated annealing proposed by Bandyopadhyay et al. [33] (AMOSA), hereinafter named as MOGA + SA (see Algorithm 1). Moreover, two different schemes from the literature have been assessed NSGAII [26] and SPEA2+ [27]. The same chromosome structure is used in these three MOGAs. This chromosome has as many genes as incoming vessels (|𝑉|). Each gene consists of three values (see Figure 6): (1)the ID of the next vessel to dispatch (𝑖); (2)the number of QCs assigned (𝑞𝑖); (3)the buffer size after this vessel (𝑏𝑖). Itshouldbenotedthateachgenemustbecomposedof feasiblevalueswithrespecttovessel𝑖. That is, according to the problem constraints, each vessel 𝑖canbeassignedatmost QC+ 𝑖cranes. Therefore, 1≤𝑞𝑖≤QC+ 𝑖. Likewise, if the berth length is 𝐿,then𝜂𝑖≤𝑝𝑖≤𝐿−𝑙𝑖−𝜂𝑖. In the following subsections, genetic operators that are used by the implementations of NSGAII and SPEA2+ are described. 6.1. Decoding and Evaluation of One Chromosome/Solution. The structure of the chromosome, specifically the order of the vessels, is used as a dispatching rule. Hence, we use the following decoding algorithm: the genes are visited from left to right in the chromosome sequence. For each gene (𝑖,𝑞𝑖,and 𝑏𝑖), the vessel 𝑖is scheduled at the earliest mooring time with 𝑞𝑖consecutive QCs available, so that none of the constraints are violated. In case there are several positions available at the earliest mooring time, the one closest to the berth extremes is selected. After the departure of the vessel 𝑖(𝑑𝑖), it is ensured that there are 𝑏𝑖time units where no other vessel 𝑗(∀𝑗∈𝑉,𝑗 = 𝑖) uses the QCs assigned to vessel 𝑖nor moors where vessel 𝑖 does [𝑝𝑖,𝑝𝑖+𝑙𝑖). Once a valid mooring time (𝑚𝑖) and initial position (𝑝𝑖)havebeenassignedtoeachvessel𝑖,thefitnessof the chromosome (equation (16))isobtainedbycomputing each one of the objective functions: total service time ( 𝑇𝑠), robustness ( 𝑅),andstandarddeviationoftherobustness(𝜎). 6.2. Generation of Initial Population. Construction of initial population (𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛procedure) is performed so that the service time of a percentage of the initial population (GA parameter) is at least as good as the solution provided by the FCFS policy. The other chromosomes (or solutions) are constructed by instantiating each gene in the following way. (i) Vessel identifier (𝑖): an integer, between 1 and 𝑁,is randomly chosen. Two genes of the same chromosome cannot have the same vessel identifier. (ii) Number of QCs (𝑞𝑖): an integer, between 1 and QC+ 𝑖, is randomly chosen. (iii) Buffer size (𝑏𝑖):theinitialbuffersizeis0forallgenes of the initial population. Once all chromosomes in the initial population have been instantiated, their fitness values are obtained as described in Section 6.1. Furthermore, the Pareto front Xis updated considering all these chromosomes. Let 𝑥be a chromosome (or solution); 𝑥is added to the Pareto front Xif there is no other solution 𝑦∈Xsuch that 𝑦dominates 𝑥.If𝑥is added to X, then all solutions dominated by 𝑥are removed from X. 6.3. Evolution of One Population. In each iteration of the MOGA, a new population is built from the previous one (or the initial) by applying the genetic operators of selection, reproduction, and replacement. The proposed approach follows the scheme: (i) selection: all chromosomes in the actual population are randomly grouped into pairs; (ii) reproduction: (1)each one of these pairs is mated or not according to the crossover probability 𝑃𝑐 generating two offspring; (2)each offspring, or parent if the parents were not mated, undergoes mutation in accordance with the mutation probability 𝑃𝑚; (iii) replacement: after evaluating the chromosomes previously generated, a tournament selection (4 : 2) is carried out among each pair of parents and their offspring as a replacement. 6.4. Crossover. Thecrossoveroperatorreceivesonepairof chromosomes (𝑃1and 𝑃2), which are in the current population 𝑝𝑜𝑝andhavebeenrandomlyselected.Theobjective of this operator is to construct two offspring chromosomes (𝑂1and 𝑂2). For that, each time the crossover operation is performed, the following steps are made. (1) Two cross points are randomly chosen, 𝑘1and 𝑘2(1≤ 𝑘1<𝑘2≤𝑁). (2) Each gene in chromosomes 𝑃1and 𝑃2which is in position 𝑝,𝑘1≤𝑝<𝑘2,iscopiedtothesameposition in chromosomes 𝑂1and 𝑂2,respectively. (3) Each gene in chromosomes 𝑃1and 𝑃2which is in position 𝑝,1≤𝑝<𝑘1,iscopiedtothesameposition in chromosomes 𝑂1and 𝑂2,respectively. (4) Each gene in chromosomes 𝑃1and 𝑃2which is in position 𝑝,𝑘2≤𝑝≤𝑁,iscopiedtothesameposition in chromosomes 𝑂1and 𝑂2,respectively. Figure 7is a graphical representation of the procedure that is used to perform the crossover operation, which is basedonthetechniquegeneralizedpositioncrossover[34] that is commonly used in permutation based encodings. In one chromosome there cannot be two genes with the same vessel identifier. Therefore, if the vessel identifier in the gene that will be copied already exists in the offspring (𝑂1/𝑂2) 16 Mathematical Problems in Engineering Table 4: Percentages of incidences absorbed in schedules of 100 vessels. (a) Delays applied to schedules with different levels of  𝑅 Range 𝑅min 𝑅𝑖𝑅Max 𝑑∈[1,0.2ℎ𝑖]21.78 95.51 99.95 𝑑∈[1,0.5ℎ𝑖]18.73 93.58 99.85 𝑑∈[1,0.8ℎ𝑖]16.64 90.49 98.96 𝑑∈[1,1.0ℎ𝑖]13.92 87.10 98.01 𝑑∈[1,1.2ℎ𝑖]12.93 85.31 97.15 (b) Delays applied to schedules with different levels of  𝑇𝑠 Range 𝑇𝑠min 𝑇𝑠𝑖𝑇𝑠Max 𝑑∈[1,0.2ℎ𝑖]20.17 79.91 99.94 𝑑∈[1,0.5ℎ𝑖]16.16 73.88 99.75 𝑑∈[1,0.8ℎ𝑖]13.89 65.17 98.94 𝑑∈[1,1.0ℎ𝑖]11.85 60.25 98.08 𝑑∈[1,1.2ℎ𝑖]10.32 56.76 96.74 (c) Delays applied to schedules with different levels of 𝜎( 𝑅) Range 𝜎(𝑅)min 𝜎(𝑅)𝑖𝜎(𝑅)Max 𝑑∈[1,0.2ℎ𝑖]99.96 75.48 61.85 𝑑∈[1,0.5ℎ𝑖]99.95 72.23 54.42 𝑑∈[1,0.8ℎ𝑖]99.34 67.88 48.72 𝑑∈[1,1.0ℎ𝑖]98.87 65.34 47.77 𝑑∈[1,1.2ℎ𝑖]97.39 62.38 45.75 Table 5: Percentages of incidences absorbed in schedules of 100 vessels obtained using or not LS (timeout 30 secs). Range 𝑅Max no LS 𝑅Max with LS 𝑑∈[1,0.2ℎ𝑖]99.53 99.88 𝑑∈[1,0.5ℎ𝑖]99.40 99.70 𝑑∈[1,0.8ℎ𝑖]97.62 98.51 𝑑∈[1,1.0ℎ𝑖]95.33 97.34 𝑑∈[1,1.2ℎ𝑖]94.04 96.00 and thus the third objective managed in this way is to minimize the standard deviation of the robustness measurements. In this paper, a mixed integer programming (MIP) model and a new hybrid multiobjective genetic algorithm (MOGA + SA) were developed for the dynamic and continuous robust BAP + QCAP. They were compared with two well-known multiobjective genetic algorithms (MOGAs): NSGAII and SPEA2+. In multiobjective optimization problems there is no a unique optimal solution, and it is necessary to assess the trade-off between all the objectives by using the Pareto front. Visualizing Pareto fronts provides container terminals operators with a helpful system to decide which schedule is better depending on the actual state of the container terminal. The results showed that the MIP model was able to obtain robust and efficient schedules up to 10 incoming vessels. However, MOGA + SA achieved better Pareto fronts than the MIP model for queues of incoming vessels greater than or equal to 20 vessels. Thereby, the schedules obtained by MOGA + SA were more efficient and robust than the schedules obtained by the MIP model. Furthermore, the MIP model was unable to find any solution with a given timeout for a queue of 50 incoming vessels. Additionally, differences between the MOGAs have been assessed by means of nonparametric statistical tests. It turned out to be that MOGA + SA obtained better Pareto fronts according to the hypervolume measures. Furthermore, different sets of incidences were simulated into the schedules obtained by the NSGAII and the MOGA + SA. The results returned that the schedules obtained by MOGA+SA were more robust due to the fact that they could absorb more incidences. 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