Container Terminal Berth Planning

Container Terminal Berth Planning Critical Review of Research Approaches and Practical Challenges S. Theofanis, M. Boile, and M. M. Golias A comparative, up-to-date, critical review of existing research efforts relating to berth planning is presented. Existing models are critically reviewed on the basis of their efficiency in addressing key operational and tactical questions relating to vessel service and their relevance and applicability to different marine terminal operator berth-planning strategies and contractual service arrangements between terminal operators and ocean carriers. The strengths and deficiencies of the existing models in addressing real-world problems in a systemic and coherent manner are discussed. The paper concludes with some insights for future research issues. Container terminals are open systems of material flow with three distinguishable areas: the berth area, where vessels are berthed for service; the storage yard area, where containers are stored as they temporarily wait to be exported or imported; and the terminal receipt and delivery gate area, which connects the container terminal to the hinterland. Operations in a container terminal can be broken down into three functional systems: seaside operations consist of the vessels berthing operations at the quay, the unloading of containers from the vessel, and the loading of containers onto the vessel. The seaside operations interact with the yard operations through the internal transport equipment used to move containers from the vessel to the storage yard and from the storage yard to the vessel. The yard operations manage the containers during the transfer between the land side and the seaside. These operations include functions such as the internal transport of the containers from the vessel to the trucks or rail and from the trucks or rail to the vessel and the storage operations in the storage yard. The land-side operations deal with activities of receiving and delivering inbound and outbound containers to and from the storage yard. Although each operation can be viewed as an independent entity and its functions are usually studied as such, interactions between the systems are unavoidable and play a crucial role in the efficient management and operation of a container terminal. Even though seaside operations are interrelated with the container transfer and storage yard operations, terminal operators S. Theofanis, Freight and Maritime Program, Center for Advanced Infrastructure and Transportation, Rutgers University, 100 Brett Road, Piscataway, NJ 08854. M. Boile, Department of Civil and Environmental Engineering and Freight and Maritime Program, Center for Advanced Infrastructure and Transportation, Rutgers University, 623 Bowser Road, Piscataway, NJ 08854. M. M. Golias, Department of Civil Engineering, University of Memphis, 3815 Central Avenue, Memphis, TN 38152. Corresponding author: S. Theofanis, stheofan@rci.rutgers.edu. Transportation Research Record: Journal of the Transportation Research Board, No. 2100, Transportation Research Board of the National Academies, Washington, D.C., 2009, pp. 22 28. DOI: 10.3141/2100-03 have a special interest in seaside operations because of their relationship with ocean carriers. The tremendous increase in containerized trade in recent years, the resulting congestion in container terminals worldwide, the remarkable increase in containership capacity, the increased operating cost of container vessels, and the adoption by liner shipping of yield management techniques strain the relationships between ocean carriers and container terminal operators. Shipping lines want their vessels to be served immediately on arrival or according to a favorable priority pattern and complete their loading or unloading operations within a prearranged time window, irrespective of the problems and the shortage of resources that terminal operators are facing. Therefore, in many cases allocation of the scarce berthing resources is considered to be a problem deserving both practical and theoretical attention. A few published papers focused on reviewing marine terminal operations, including the berth planning-related literature (1 4). Those reviews, however, did not exclusively and extensively cover the berth-planning literature. This paper presents a critical, up-to-date review of the existing research efforts relating to the berth allocation problem (BAP). The existing models have been critically reviewed on the basis of their efficiency in addressing key operational and tactical questions relating to vessel service and their relevance and applicability to the different strategies and contractual service arrangements between terminal operators and shipping lines. The strengths and limitations of the existing models for addressing realworld problems in a systemic and coherent manner are discussed. Research efforts simultaneously addressing BAP and the quay crane (QC) assignment (QCA) problem are not covered. The paper concludes with a critical overview of the issues to be addressed to make these models more relevant to real-world applications and some insights for future research issues. BERTH ALLOCATION PROBLEM The BAP can be simply described as the problem of allocating berth space for vessels in a container terminal. Vessels usually arrive over time, and the terminal operator needs to assign them to berths to be served (to unload and load containers) as soon as possible. Ocean carriers and therefore vessels compete over the available berths, and different factors, discussed in detail later in this paper, affect the berth and time assignment. BAP has three planning and control levels: the strategic, the tactical, and the operational levels. At the strategic level, the number and the lengths of the berths and quays that should be available at the port are determined. This is done either during the initial development of the port or when an expansion is considered. At the tactical level, midterm decisions are usually taken, for example, 22

Theofanis, Boile, and Golias 23 the exclusive allocation of a group of berths to a certain ocean carrier. At the operational level, the allocation of berthing space to a set of vessels scheduled to call at the port within a time horizon of a few days must be decided on. Normally, this planning horizon does not exceed 7 to 10 days. Because ocean carrier vessels follow a regular schedule, in most cases the assignment of a berth to the vessel must be decided on a regular and usually periodical basis. At the operational level, the BAP is typically formulated as a combinatorial optimization problem. After the BAP has been solved, the resulting berth-scheduling plan is usually presented by the use of a time space diagram. The BAP has been formulated according to the following variations: (a) discrete versus continuous berthing space, (b) static versus dynamic vessel arrivals, and (c) static versus dynamic service time. The BAP can be modeled as a discrete problem if the quay is viewed as a finite set of berths, in which each berth is described by fixed-length segments or points. Because vessels are of different lengths, dividing the quay into a set of segments is difficult, mainly because of the dynamic change of the length requirements for each vessel. One solution to this problem is the use of longer segments (a solution that results in the poor use of space) or short segments (an approach that leads to infeasible solutions). To overcome these drawbacks, continuous models have appeared in the literature. In these models vessels can berth anywhere along the quay. In the former case, the BAP can be modeled as an unrelated parallel machine-scheduling problem (5), whereas in the latter case the BAP can be modeled as a packaging problem. The BAP can also be modeled as a static BAP (SBAP), if all the vessels to be served are already in the port at the time that scheduling begins, or as a dynamic BAP (DBAP), if all the vessels to be scheduled for berthing have not yet arrived but their arrival times are known in advance. The service time at each berth depends on several factors, with the two most important being the number of cranes operating at each vessel and the distance from the preferred berthing position, that is, from the berth with the minimum distance from the storage yard blocks, where containers to be loaded onto or unloaded from the vessel are stored. If the model does not take into consideration the number of cranes operating at each vessel, the problem can be considered static in terms of the handling time. But if this number is decided on from the model, the formulation can be considered dynamic in terms of the vessel service time. Finally, technical restrictions such as berthing draft and intervessel and endberth clearance distances are further assumptions that have been considered. The model formulations that have appeared in the literature combine two or more of these assumptions and, in most cases, lead to nondeterministic polynomial-time hard problems that require the development of heuristic algorithms to achieve computationally acceptable solution times. LITERATURE REVIEW The first paper on BAP to appear focused on queuing theory, and the study used a serial representation of the problem (6). Issues relating to the applicability of this modeling approach to BAP were discussed elsewhere (7). After 1990, research on BAP focused solely on mathematical programming and simulation, because queuing models failed to capture several of the attributes of the problem, mainly the space attribute associated with container terminal berthing. The only exceptions were a study by Legato and Mazza (8) and a study by Dragovic et al. (9). The latter investigators used queuing theory and simulation to evaluate the efficiency of the models used at the Pusan East Container Terminal, whereas the former investigators presented a closed queuing network model and performed a simulation for estimation of the effects of congestion at the berth. A recent study also used a queuing network model for berth allocation and crane assignment (10). One of the early works that appeared in the literature and that did not follow a serial approach was by Lai and Shih (11). The authors assumed that a wharf was partitioned into several sections, to each of which only one vessel could be allocated at a specific time. The berth allocation rules considered the following factors: the available sections of a wharf, the expected completion time for a vessel at each available section, and the size and the arrival time of each vessel. A heuristic algorithm was developed by considering a first-come first-served (FCFS) rule, and simulation experiments compared three berth allocation rules. The BAP in naval ports was treated (12, 13), and the optimal set of vessel-to-berth assignments that maximized the sum of benefits for the vessels while they were in port was identified. The idea that for high port throughput, optimal vessel-to-berth assignments should be found without considering the FCFS basis was introduced (14). This approach may result in some vessels dissatisfaction regarding the order of service. To deal with these two conflicting criteria, a heuristic algorithm was developed with the aim of finding the set of noninferior solutions. The two-objective problem was formulated as a single-objective problem. Numerical experiments showed that the trade-off between the two objectives increased with the size of the port. That paper was the first one that specifically considered customer satisfaction but did so in an aggregate way. The continuous-space SBAP (CSBAP) was formulated as a scheduling problem with a single processor through which multiple jobs can be processed simultaneously, and the minimization of the makespan was attempted on the basis of that assumption (15). A first-fitdecreasing heuristic rule was suggested, and numerical experiments were conducted for that rule. The results showed that the average relative errors of the heuristics were less than 20%. Similar to Li et al. (15), Guan et al. (16) considered the BAP to be a multiprocessor task-scheduling problem. A vessel was assigned to a number of QCs, and a heuristic to minimize the total weighted completion time was proposed. A worst-case analysis was performed to evaluate the proposed heuristic. Weights, which were dependent on the vessel size, were assigned to each job, and this method can be considered a form of implicit rule implementation. Similar to Li et al. (15), CSBAP was addressed by Lim (17) with the objective of minimizing the maximum amount of quay space used at any time. The approach was very restricting because it did not consider the berthing time to be a decision variable and the handling time did not vary along the quay. The problem was represented as a graph with directed and undirected edges and was transformed into a restricted version of the two-dimensional packing problem. A heuristic that performed well with historical data was presented. The discrete-space DBAP (DDBAP) was addressed with the objective of minimizing the sum of the vessel s waiting and handling times (18). A Lagrangian relaxation heuristic was proposed to solve the resulting problem. The heuristic performed well from a practical point of view. Similar to other work (15, 16), Park and Kim considered CDBAP and used a subgradient optimization technique with the objective of minimizing the total waiting and service time and the deviation from the preferred berthing location (19). They were the first to include penalization of the deviation from the optimal berth. The implementation of the ant colony optimization technique was presented by Tong et al., who showed how it can be effectively applied to solve

24 Transportation Research Record 2100 the continuous BAP (20). The objective was to minimize the necessary wharf length, subject to several space and time constraints. Experimental results were presented, but there was no indication that the algorithm may perform well for real-life problems. In the same context of the work of Imai et al. (18), Nishimura et al. (21) addressed the same problem with the same objective but for a public berth system, including physical restrictions and dropping the assumption that Imai et al. (18) used that each berth can handle one vessel at a time. Service priority relied on the FCFS rule and was not dependent on the vessel s cargo volume. A heuristic based on genetic algorithms (GAs) was used, and experimental results were presented for both the simultaneous and the single occupancy of a berth. The model formulation presented by Imai et al. (18) was criticized by Hansen and Oguz (22), and a new formulation was presented. A corrigendum was published to clarify this issue (23). To further investigate this issue, several computational experiments were performed (24) by using the formulations presented previously (18, 22). In all test instances, the same optimal objective function value was obtained. It should be noted that because of the multisolution space of the BAP, different assignments will provide the same optimal objective value. Imai et al. (25) extended and modified the DBAP formulation (18, 21) to include service priority constraints by the use of weights. The objective was to minimize the total weighted service time. Several numerical examples were presented by using different weight priority formulas, and a brief but non-in-depth discussion on the choice of the value for the weights was provided. Kim and Moon studied CDBAP (26). They presented a mixedinteger problem (MIP) formulation and used simulated annealing as a solution approach to minimize the delays and the handling costs resulting from nonoptimal locations of vessel berths. Priority service rules were implicitly included with a late departure penalty. A comparison of the simulated annealing method with the classical optimization technique was also presented for small instances of the problem. The results showed that for these instances the simulated annealing method provided nearly optimal solutions within a reasonable time frame. The previous work (26) was extended (27) with the consideration of QC capacities. The study determined the optimal start times and mooring locations of vessels and determined the optimal assignment of QCs to each vessel. Handling times were assumed to vary linearly with the number of QCs assigned to each vessel. The handling time was considered to be independent from the berthing location, which is a limitation. The formulation of the objective function, similar to that presented elsewhere (19, 26), included four different cost penalties (handling cost, early arrival, late arrival, and late delay). Similar to the work of Park and Kim (27), Miesel and Bierwirth (28) presented a model for the combined BAP and QCA to minimize the cost of operating QCs. A priority rule-based method was used to determine the mode, the berthing time, and the berthing position of each vessel. On the basis of real-world data, the proposed approach considerably improved the berth plans without deteriorating the service quality. Similar to the other work (27, 28), Lee et al. (29) presented a method for the DDBAP and QCA to minimize the total service time and make-span of all the vessels and the completion time for the QCs. A GA-based heuristic was proposed as a solution approach. The heuristic performed well for small instances of the problem. An unpublished paper described the study of both CSBAP and CDBAP as a rectangle-packing problem with arrival constraints (30). The objective was to minimize the delays of vessels, with higherpriority vessels receiving guaranteed levels of service, and at the same time address the desirability of berthing vessels at designated locations to minimize the movements of containers within the yard and between the vessels. The minimum-total-weighted-flow-time CDBAP was studied (31). Two formulations were considered, and according to an idea presented earlier (25), weight coefficients were applied to the vessels (although application of the coefficients was limited to vessels departing later than a requested time). A tree procedure was developed along with a heuristic that combines the tree procedure with the heuristic presented previously (16). Computational experiments showed that the composite heuristic was quite effective. Imai et al. (32) extended the previous work (14, 18, 21, 23, 25) by solving the CDBAP. They assumed that the handling time depends on the quay location where the vessel is handled and that it is a function of the berth location relative to the container storage yard and the assigned yard trailers that transport containers to or from the vessel. The objective was restrained, as with most of the papers presented to date, to the minimization of the total service and wait time for all the vessels. Similar to the work of Imai et al. (18), the DDBAP was considered (33), and two formulations were provided: a formulation similar to that of Imai et al. (18) and a formulation similar to that of the multidepot vehicle routing problem with time windows, which minimizes the total weighted service time. The authors also developed a heuristic for the continuous case (33). In contrast to previous models, their work can handle a weighted sum of service times as well as windows on berthing times. Medium-sized instances of the DDBAP were solved exactly under some assumptions, which provided an assessment of the quality of the heuristic. Because the continuous problem could not be solved exactly, the assessment of the heuristic developed was inferred only from the discrete scenario. The integration between a flexible simulator representing the marine-side operations of a container terminal and a linear programming model for improving berth assignment and yard stacking management policies was outlined (34). An artificial intelligence approach was introduced; the assignment of vessels to berths was based on several factors, including minimization of the waiting time of the vessel, berth productivity, and the minimum distance for vessel berthing and sailing (35). The results showed a reduced average waiting time for vessels, whereas several other measures of port productivity were also considered. The minimum-service-time DDBAP with indented berths was addressed but with the constraint that megavessels were to be served without delays (36). A GA heuristic was used to solve the problem, and the computational results showed that megavessels were served faster at the expense of the total service time. A variation of the DDBAP, in which vessels exceeding a wait time limit were assigned to an external terminal, was addressed (37). The objective of the problem was to minimize the total service time of vessels at the external terminal. The CDBAP was solved with the objective of minimizing the unallocation, position, and delay costs by using a stochastic beam search heuristic (38). The authors concluded that the formulation and solution approach is fast, can easily be modified and implemented, and can be directly applied to solving multistage decision-making problems. A linear reformulation of the DDBAP proposed by Imai et al. (25) was presented (39), and a rolling-time-window heuristic for solving medium and large instances of the problem was proposed. Computational examples and a discussion of the efficiency of the heuristic were not provided. The DDBAP was formulated as a linear MIP with the objective of simultaneously minimizing the cost from late departures and maximizing the benefits from early departures (40). A solution approach was not presented, and computational examples were limited to the conceptual benefits of the proposed approach. The authors showed that the proposed formulation could be reduced to the DDBAP formulations presented elsewhere (18, 25). The work by Golias et al.

Theofanis, Boile, and Golias 25 (40) was extended (41), and a formulation for the DDBAP reflecting time-window service deadlines was presented. The problem was formulated as a linear MIP with the objective of simultaneously minimizing the cost from vessel late departures and maximize the benefits from vessel early and timely departures. The authors considered only the premiums and penalties from early departures. The first study to present a multiobjective optimization and resolution approach for the DDBAP was by Boile et al. (42). The objective was to minimize simultaneously the total service time for all vessels and the delayed departure and berthing costs of preferential customers by using two separate objective functions. A DDBAP considering the minimization of total costs for waiting and handling as well as the earliness or the tardiness of completion for all vessels was studied (43). The formulation presented can be reduced to the DDBAP presented by Imai et al. (18, 25). The reduction of the model was simpler than the one presented earlier (40, 42). A variable neighborhood search (VNS) heuristic was proposed and compared with the multistart heuristic, GA heuristic, and memetic search heuristic. The VNS heuristic provided optimal solutions for small instances but only for the minimum total service time problem. The work of Dai et al. (30) was expanded by Moorthy and Teo (44), who presented a new approach for the CDBAP with stochastic vessel arrivals and who achieved promising results. The problem was modeled as a bicriterion optimization problem. The first objective dealt with the trade-off between the operational cost of moving containers from one vessel to the other and the delays of the start of berthing. The second objective dealt with the robustness of the final schedule given stochastic vessel arrivals. To the best of the authors knowledge, that is the first study to have solved a berth template problem and analyzed the impact of the template on the real-time berth allocation. A Lagrangian heuristic was developed to solve the minimum-service-time DDBAP (45). An adaptive local search heuristic algorithm for the DDBAP was also proposed (46), while a post Pareto analysis for the multiobjective DDBAP was presented (47). A variable neighborhood heuristic for the minimum-service DDBAP was proposed (48). The work by Boile et al. (42) was extended by Golias et al. (49) to include customer differentiation. A nonlinear modeling approach for the berthscheduling problem was presented (50), and two-solution algorithms based on GAs were proposed. A biobjective formulation for the DDBAP to minimize vessels delays and total service time was presented (51). A GA and a subgradient optimization-based heuristic were proposed to solve the resulting problem. A formulation between the DDBAP and the CDBAP was proposed (52). A neighborhood search-based approach was proposed, and utility indexes were used to provide priorities among vessels by maximizing these utilities. The novelty of this approach was that the central processing unit could handle large-scale problems within a short time. Following previous work (14, 25, 49), the concept of hierarchical optimization to prioritize between the different objectives of the BAP was introduced (53, 54). Another recently presented approach (Golias, M. M., G. Saharidis, M. Boile, S. Theofanis, and M. Ierapetritou, unpublished work; 55) proposed a berth-scheduling policy (BSP) in which vessel arrival times were considered a variable and were optimized. Finally, a second and novel attempt to capture the stochastic nature of the BAP was proposed, in which vessel arrival and handling times were considered stochastic parameters of the problem (56). The objective focused on the minimization of the expected waiting time for all the vessels. A GA-based heuristic was applied to solve the problem, although the computational examples were limited. Table 1 summarizes information for a selection of the literature presented earlier. The first column states the reference number, as it appears in the reference list at the end of this paper. The BAP variation considered, the objective and the problem formulations, and information on the solution approach adopted are provided. DISCUSSION OF CURRENT BERTH ALLOCATION APPROACHES Port Operator Service Agreements and Berth Allocation Models The SBAP and the DBAP, along with the discrete and continuousspace BAPs, have been widely studied in different combinations. Most of the studies minimized the total service time or the deviation from the preferred berth because it is expected that minimization of the deviation from the preferred berthing position will reduce the service time and the operator s cost, whereas few studies incorporated the minimization of costs endured by late departures. These objectives satisfy most of the port operators objectives but fail to portray most of the service priority agreements. These contractual arrangements, which greatly influence the different BSPs, can vary from berthing and the start of the cargohandling operations on arrival to guaranteed service time windows and guaranteed service productivity. The earliness or lateness of the start time or the time of completion of the handling operations for a vessel (i.e., the loading or unloading of containers) implies benefits or costs to both the port operator and the ocean carrier. If these operations are completed after a specified and agreed-upon time, the port operator may pay a penalty to the ocean carrier, whereas if these operations are completed before that time, the carrier may pay a premium fee (an early premium) to the port operator, subject to the contractual arrangements, although in practice premiums may be compensated for by past or future penalties assigned to the port operator because of failure to meet the terms of the contract. Although early departures are seldom reported to happen, they can help ocean carriers manage the time factor of their service schedules by providing a time buffer to cope with time lost in other ports (57). Early premiums can be offset by reducing the voyage operating cost by reducing the voyage speed (i.e., reducing fuel consumption) in the subsequent trip leg. In fact, recently, ocean carriers have sought to reduce operating costs or cope with container slot overcapacity through the use of a voyage speed reduction, the so-called slow steaming, while maintaining service punctuality. It was not until 2002 that researchers began to recognize the significance of penalties for late departures and premiums for early departures (19, 27, 31, 40, 43, 51). Nevertheless, service deadlines (the start or finish time of service) in the form of time windows, penalties, and premiums from the early or late start of service and premiums from the start of the completion of service within the deadline time window have not yet been thoroughly investigated, although they represent one of the most basic decision-making issues in real-world container terminal operations. A line of research that has not yet been fully exploited is that of combining different service agreements for different customers, including the optimization of the vessel arrivals to minimize the environmental impacts from vessel emissions during idle at the port and maximize the vessel s operational characteristics while in transit to the port (Golias, M. M., G. Saharidis, M. Boile, S. Theofanis, and M. Ierapetritou, unpublished work; 55, 58 60).

26 Transportation Research Record 2100 TABLE 1 Different Berth Planning Assignment Problem Assumptions Paper or Author BSF Objective Formulation Solution Approach 15 SCQ Makespan Single processor scheduling Heuristic 17 SC Amount of quay 2-D packaging Heuristic 16 SCQ Total weighted completion time Multiprocessor scheduling Heuristic 31 SC Total weighted completion time MIP Heuristic 19 SC Cost from delayed departures and cost of nonpreferred berth MIP Lagrangean relaxation, subgradient optimization 26 SC Cost from delayed departures and cost of nonpreferred berth, MIP Simulated annealing cost from early or late start of vessel handling against ETA (estimated time of arrival) 27 DC Handling cost, penalties from berthing before or after ETA, MIP Lagrangean relaxation, penalties from late departures subgradient optimization, dynamic programming 21 DD Total completion time MIP GA-based heuristic 18 DD Total completion time MIP Lagrangean relaxation and subgradient optimization 25 DD Weighted total completion time QAP GAs based heuristic 32 DC Total completion time MIP Heuristic 36 DC Total completion time MIP GA-based heuristic 37 DC External berth service time MIP GA-based heuristic 33 DC Total completion time/weighted total completion time MIP Tabu search heuristic 34 SC Cost from delayed departures and cost of nonpreferred berth MIP Heuristic 29 SCQ Makespan and quay crane time MIP GA-based heuristic 38 SC Position, delay, and unallocation cost MIP Stochastic beam search 43 DD Delayed departures, waiting and handling cost MIP Variable neighborhood search 44 DC Operational cost and maximize service levels Rectangle packing Simulated annealing 41 DD Delays from late and premiums from early/timely departures MIP GA-based heuristic 39 DD Multiobjective: total service time and preferred customers MIP GA-based heuristic satisfaction 49 DD Biobjective: total service time and delayed departures MIP Subgradient, GAs 51 DD Hierarchical: conceptual Bilevel MIP GA-based heuristic, exact, combinations NOTE: BSF = berth scheduling formulation; SC = static arrival time and continuous space; Q = quay crane consideration; DC = dynamic arrival time continuous space; DD = dynamic arrival time and discrete space; QAP = quadratic assignment problem; MIP = mixed integer problem. Service Priority Schemes on the Basis of Weights The allocation of vessels to berths by simply minimizing the total completion time can lead to problems, because vessels with smaller handling volumes receive higher priorities than vessels with larger handling volumes (5, 22, 25). The latter vessels end up being served at the end of the queues at each berth. Therefore, given two vessels with different handling volumes that arrive at the same time, the large vessel will wait for the smaller vessel to finish service (if they are served at the same berth). Assignment policies based on this objective have the consequence of longer waiting times for larger vessels. For a number of reasons (mainliner service, etc.), however, some large vessels might need to be assigned or finish their service as soon as possible after their arrival at the port. In practice, the problem of assigning priority status to vessels is more complex, because for some vessels signed contractual agreements between ports and customers do not allow for arbitrary assignment (30). A number of studies have attempted to address this issue by assigning weights to represent the vessels priorities. Service priority schemes based on the assignment of arbitrary weights to vessels were introduced (25) and were adopted by a number of research efforts that followed. The main issue with this approach is the determination of weights because there is no intelligent way, excluding the use of iterative processes, to assign these weights to meet specific contractual agreements. To avoid these issues, monetary penalization (26, 27) or premiums for delayed or for early or timely departures (38, 40, 41, 43) have been used. Another approach that avoids the assignment of monetary values for the tardiness or earliness of service deadlines was developed (42, 53). By that approach, different objective functions were simultaneously used to portray the different priorities of customers and the service level of the berth schedule. The advantage of this approach is the ability to capture the multiobjective and hierarchical nature of the BAP and present the port operator with a variety of possible solutions while using time as a measure of customer satisfaction and avoiding the limitations of the weighted approach. Public-Use Versus Dedicated-Use Terminals The berth allocation modeling approaches reviewed here seek to schedule vessels to berths in public-use container terminals in

Theofanis, Boile, and Golias 27 an optimal manner, although this may not have been explicitly stated by all the researchers, without losing their applicability to the dedicated-use-type container terminals. In the latter case, when terminals are owned or leased and operated by an ocean carrier, the vessels belonging to that ocean carrier may have priority over the vessels belonging to other ocean carriers calling at the terminal or the terminal may be exclusively used by the vessels of the shipping group operating it. Nevertheless, and excluding the policy of minimization of the total service time for all the vessels, the rest of the approaches are applicable and can capture the policies implemented at dedicated-use terminals. The latter statement especially applies to modeling approaches with single-level or multilevel, multiobjective formulations. These formulations are more appropriate for optimization problems with conflicting objectives and can easily incorporate the different berthscheduling objectives of dedicated terminals (42, 49, 51, 53, 54). From a practical point of view, multiobjective formulations (single or multilevel) can adequately capture the practical constraints that dedicated marine terminal operators face and deal with the preferential treatment given to certain customers or classes of vessels belonging to the same ocean carrier (e.g., the terminal operator s shipping group vessels are given preference over other ocean carriers vessels, or mainliner vessels are given preference over feeder vessels). Formulations that use weights or utility functions (25, 43, 52) can also be applied, although they are less flexible. In the latter formulations, defining the values of the weights or the values of the utility functions that would guarantee the desired vessel priorities (in terms of either the service order or punctual berthing and departure) can become a time-consuming iterative task. The introduction of soft constraints (e.g., a service time window) for certain vessels can be also applied to dedicated-use terminals. For instance, the mainliner service vessels of the operator s shipping group may be treated as priority customers. Computational Example Data Set Selection The authors would like to emphasize the importance of selecting computationally challenging data sets to evaluate the different policies and model formulations and, more importantly, test the effectiveness of the proposed resolution algorithms. For example, if real-world data are available, the range of variables and parameters of the example data sets should be based on the peak periods of the terminal s operations. The testing of solution algorithms for small examples (i.e., the DDBAP with five berths and 10 vessels) is of little value when the behavior of a policy or a solution algorithm is evaluated. As a general guideline, the authors propose the use of the parameter total volume/(berth/day) for identification of the size of the experimental data sets, along with vessel interarrival times. The use of common measures to express the level of computational complexity of experimental data sets would also provide a straightforward measure for comparison of the efficiencies of the different solution algorithms. The authors would also like to point out that although berth scheduling is an operational problem, the computational efficiency of the solution algorithm is not as crucial as it is for other operational models (i.e., internal transport vehicle routing), as long as it is in the range of minutes. Terminal operators create berth schedules ahead of time (a minimum of 1 day before the new berth schedule goes into effect), and thus, solution algorithms that can provide the required berth schedule within 1 or 2 h can be considered adequate. CONCLUSIONS The BAP has received much attention by the research community during the last 15 years. Researchers have adopted a broad spectrum of optimization objectives and solution approaches to make this particularly complex problem tractable. Recently, researchers have attempted to capture real-life berth-planning issues relating to the relationship of the marine terminal operators with the ocean carriers, particularly as far as service agreements between them are concerned, in their modeling formulations. The extent to which these efforts can be practically exploited remains to be seen. 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