Integrated Scheduling of Vessels, Cranes and Trains to Minimize Delays in a Seaport Container Terminal

Citation: Muñuzuri, J.; Lorenzo- Espejo, A.; Pegado-Bardayo, A.; Escudero-Santana, A. Integrated Scheduling of Vessels, Cranes and Trains to Minimize Delays in a Seaport Container Terminal. J. Mar. Sci. Eng. 2022,10, 1506. https:// doi.org/10.3390/jmse10101506 Academic Editor: Claudio Ferrari, Nam Kyu Park and Kevin X Li Received: 30 September 2022 Accepted: 14 October 2022 Published: 16 October 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Journal of Marine Science and Engineering Article Integrated Scheduling of Vessels, Cranes and Trains to Minimize Delays in a Seaport Container Terminal Jesús Muñuzuri * , Antonio Lorenzo-Espejo , Ana Pegado-Bardayo and Alejandro Escudero-Santana Departamento de Organización Industrial y Gestión de Empresas II, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Cm. de los Descubrimientos, s/n, 41092 Seville, Spain *Correspondence: [email protected]; Tel.: +34-954487205; Fax: +34-954487248 Abstract: The multiple processes taking place on a daily basis at an intermodal container terminal are often considered individually, given the complexity of their joint consideration. Nevertheless, the integrated planning and scheduling of operations in an intermodal terminal, including the arrivals and departures of trains and vessels, is a very relevant topic for terminal managers, which can benefit from the application of Operations Research (OR) techniques to obtain near-optimal solutions without excessive computational cost. Applying the functional integration technique, we present here a mathematical model for this terminal planning process, and solve it using heuristic procedures, given its complexity and size. Details on the benchmark comparison of a genetic algorithm, a simulated annealing routine and a tabu search are provided for different problem instances. Keywords: OR in maritime industry; logistics; artificial intelligence; intermodal container terminal ; transshipment 1. Introduction The representation of Figure 1is well known to researchers focusing on intermodal container terminal management. These terminals receive and dispatch containers either on vessels, trains or trucks, as well as acting as supply chain buffers where containers, empty or full, can be stored. The internal equipment of the terminal consists mainly of berth cranes, truck cranes and rail cranes, in addition to additional auxiliary cranes and internal trucks used to move containers between the berth, rail and stacking areas. Due to the rising volumes of freight moved through this type of terminals on increasingly large vessels and trains using an increasingly large number of cranes and trucks, the interest of researchers towards the management of these enormously complex systems in a globalized world has continued to grow over the last two decades. A recent systematic literature review of maritime transport research performed by Bai et al. [ 1 ] highlights “Intermodal transport” as an emerging topic in the past five years. Interestingly, said topic is found to be closely linked to “Port management”, an indicative of the surge in organizational complexity generated by employing multiple modes of transport. Several recent reviews [ 2 – 7 ] cover the main research areas and results related to container terminal planning and management. The analysis of these reviews proves that the complexity of the terminal’s systems often results in the problem’s dissection, leading researchers to focus only on one part of the overall system: berth allocation, crane scheduling, stacking and internal transport, drayage operations, etc. As a result, all the above reviews identify the integrated planning of operations as one of the main emerging areas of research on intermodal terminals. J. Mar. Sci. Eng. 2022,10, 1506. https://doi.org/10.3390/jmse10101506 https://www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2022,10, 1506 2 of 16 J. Mar. Sci. Eng. 2022, 10, x FOR PEER REVIEW 2 of 17 Figure 1. Schematic representation of an intermodal seaport terminal. Some recent research works have thus faced the problem of considering all (or most of) the main terminal operations together in the planning process. Kozan [8] applies a network model approach, solving a mixed integer linear problem to minimize the handling times of containers between their arrival and their departure from the terminal. Chen et al. [9] formulate a scheduling-based approach seeking to minimize the makespan of the processing of containers (jobs) by the different cranes and vehicles (machines), solving the problem with a tabu search routine. Alessandri et al. [10] propose the representation of the movement of containers inside the terminal by a system of queues, whose dynamic evolutions are described by discrete-time equations. In these equations, the state variables represent the queue lengths, and the control variables take into account the utilization of terminal resources, with the optimization problem consisting of the minimization of transfer delays in the terminal. Chen et al. [11] incorporate routing aspects to the planning of yard trucks, integrating them with the scheduling of cranes by means of a three-stage process. Finally, Lu and Le [12] address the stochastic nature of the scheduling problem, solving the resulting linear programming model with binary variables using a particle swarm algorithm. However, the above examples only take into account the scheduling of quay and yard cranes and yard vehicles, without focusing on the arrival and departure of vessels and trains. For instance, Kozan [8] and Lu and Le [12] do not contemplate these arrivals and departures at all, while Alessandri et al. [10] model them either as deterministic processes or as random sequences. However, other authors have also integrated the consideration of vessel and train scheduling with internal operations, assuming there exists a certain control over arrivals and departures in order to improve the efficiency of the overall transport chain. For instance, Boros et al. [13] consider the scheduling of a vessel calling at a port in relation with the container yard availability, with the point of view of establishing an optimal cycle period for the vessel. Wong and Kozan [14] integrate berth cranes with vehicles and yard storage in order to improve operation efficiency, solving the resulting model with List scheduling and tabu search algorithms. However, the VESSEL BERTH CRANES BERTH AREA RAIL CRANES RAIL TERMINAL STACKING AREA Figure 1. Schematic representation of an intermodal seaport terminal. Some recent research works have thus faced the problem of considering all (or most of) the main terminal operations together in the planning process. Kozan [ 8 ] applies a network model approach, solving a mixed integer linear problem to minimize the handling times of containers between their arrival and their departure from the terminal. Chen et al. [ 9 ] formulate a scheduling-based approach seeking to minimize the makespan of the processing of containers (jobs) by the different cranes and vehicles (machines), solving the problem with a tabu search routine. Alessandri et al. [ 10 ] propose the representation of the movement of containers inside the terminal by a system of queues, whose dynamic evolutions are described by discrete-time equations. In these equations, the state variables represent the queue lengths, and the control variables take into account the utilization of terminal resources, with the optimization problem consisting of the minimization of transfer delays in the terminal. Chen et al. [ 11 ] incorporate routing aspects to the planning of yard trucks, integrating them with the scheduling of cranes by means of a three-stage process. Finally, Lu and Le [ 12 ] address the stochastic nature of the scheduling problem, solving the resulting linear programming model with binary variables using a particle swarm algorithm. However, the above examples only take into account the scheduling of quay and yard cranes and yard vehicles, without focusing on the arrival and departure of vessels and trains. For instance, Kozan [ 8 ] and Lu and Le [ 12 ] do not contemplate these arrivals and departures at all, while Alessandri et al. [ 10 ] model them either as deterministic processes or as random sequences. However, other authors have also integrated the consideration of vessel and train scheduling with internal operations, assuming there exists a certain control over arrivals and departures in order to improve the efficiency of the overall transport chain. For instance, Boros et al. [ 13 ] consider the scheduling of a vessel calling at a port in relation with the container yard availability, with the point of view of establishing an optimal cycle period for the vessel. Wong and Kozan [ 14 ] integrate berth cranes with vehicles and yard storage in order to improve operation efficiency, solving the resulting model with List scheduling and tabu search algorithms. However, the consideration of vessels only leads J. Mar. Sci. Eng. 2022,10, 1506 3 of 16 to the assumption of predefined load/unload plans, which we have relaxed in our work. Furthermore, the minimization of service times might lead to terminal inefficiencies. Qi and Song [ 15 ] optimize the scheduling of a liner vessel following a cyclic route, which is subject to uncertainty in navigation and port processing times, whereas Vacca et al. [ 16 ] solve the integrated berth allocation problem and quay crane assignment problem (BAPQCAP) by forcing each vessel to be serviced within a specified time window. Finally, Zhang et al. [ 17 ] coordinate the scheduling of vessels with the distance to the assigned berth and the navigation through the access channel. With respect to train scheduling, the optimization of shunting yards, and areas used to separate freight trains and reassemble new trains, is addressed by Boysen et al. [ 18 ] and Jaehn et al. [ 19 ], with the objective of minimizing overall delays and waiting times. The work by Caballini et al. [ 20 ] specifically addresses the problem of the sequencing of freight trains arriving and departing at an intermodal terminal. To our knowledge, only Zhang et al. [ 21 ] include both vessels and trains in a multi-objective mathematical model to optimize the connection between both transport modes, but they do not identify the transshipment of individual container groups, as well as not bringing the scheduling of cranes into the analysis. Even if they include fewer aspects of the integral management of a container terminal, Yan et al. [ 22 ] present an integrated approach to train scheduling at an intermodal seaport with direct transshipments between vessels and trains. In fact, they posit the maximization of the number of direct transshipments as part of their objective function. Furthermore, Zhang and Li [ 23 ] consider vessel-to-train transshipments as well as multi-time window constraints and stochastic ship delays, but they only address the flow of containers loaded and unloaded at the vessels and trains, thus lacking an integral approach. Figure 2shows the sequence of optimization tasks integrated in this paper. This sequence includes the BAP, the QCAP and the quay crane scheduling problem (QCSP), the main optimization tasks identified by Bierwirth and Meisel [ 24 ], together with the rail crane scheduling problem (RCSP), addressed separately by several recent works such as by Jeong and Kim [ 25 ], Boysen et al. [ 26 ] or Guo et al. [ 27 ]. Our focus is on the integrated consideration of all these problems, including not only the scheduling of operations in the terminal but also considering that the arrival times of vessels and trains are decision variables (within certain boundaries). This corresponds to the case when the terminal is in charge of assigning slots to vessels and trains depending on a series of factors such as their arrival expectations, their priority, the path to be followed by each container within the terminal, the terminal’s equipment capacity, etc. This scenario enables a more efficient intermodal connection between sea, rail and road, seeking to reduce delays and costs, since this planning operation would be executed well in advance, and the different transportation means would be informed so that they can adjust their exact arrival times accordingly. This advantage is of greater importance in cases such as inland ports, where additional resource and workforce planning is required due to the complexity of the arrival and departures of vessels at the port [28]. Seeking to incorporate all these factors into the integrated scheduling problem, the following section describes the characteristics of the problem we focused on, which will then lead to the formulation of the resulting mathematical optimization model. Given the complexity of the problem, a heuristic approach follows, validated by comparison with exact solutions to the model and thereafter used to solve a series of larger problem instances based on the operation of the port of Seville, in southern Spain. The paper ends with conclusions and directions for future work. J. Mar. Sci. Eng. 2022,10, 1506 4 of 16 J. Mar. Sci. Eng. 2022, 10, x FOR PEER REVIEW 4 of 17 Figure 2. Sequence of planning tasks affecting operations in an intermodal terminal. 2. Problem Description Every day, intermodal container terminals receive a number of transport units (trucks, trains and vessels) loaded with containers that need to be transferred. These containers will first be unloaded and then spend some time in the container yard or, ideally, be immediately transferred to their assigned departing transport unit. In order to avoid bottlenecks and unnecessary delays, the terminal can assign time slots to each incoming transport unit, so that its total service time (the time required to unload it and then load it again and have it ready for departure) is as reduced as possible. The terminal planner knows in advance which containers will arrive on each incoming transport unit, and on which outbound transport unit they must leave the terminal. The service time required for each vessel or train depends on the availability of cranes, and different berth, rail and truck cranes can be assigned to carry out load and unload operations for vessels, trains and trucks. Several cranes may work at the same time on the same vessel or train, but under a single cycle, assumption cranes operating both on trains and vessels can only start loading outgoing containers after all the incoming containers have been previously unloaded. We have assumed that the sequence of loading and unloading operations on vessels is predefined, whereas it needs to be determined for the case of trains, but the formulation is general enough to include different configurations, and the reader may replace “vessels” with “transport units with predefined unload/load sequences” and “trains” with “transport units without predefined unload/load sequences”. Even though the modeling of the problem can have a generic application, we have included several additional assumptions to simplify its mathematical modeling. In the first place, we assume it applies to an intermodal import-export chain, so that containers arriving to the terminal on a train will be transferred to a vessel and vice versa. This does not represent a loss of generality by not considering trucks, since it is always possible to model trucks as trains that carry one single container, and the benefits of establishing slots for trucks in an intermodal terminal have been demonstrated by Zehendner and Feillet [29]. Furthermore, we have not included the stacking area in the model specifically, but it can be represented by a virtual vessel and train with unlimited capacity, arriving at the beginning of the working day and staying until the end, which absorb all the containers that go to the stacking area and emit all the containers extracted from the stacking area to be loaded on departing transport units. Finally, our concern will not be on individual containers but on container groups, sets of containers that arrive together on the same Figure 2. Sequence of planning tasks affecting operations in an intermodal terminal. 2. Problem Description Every day, intermodal container terminals receive a number of transport units (trucks, trains and vessels) loaded with containers that need to be transferred. These containers will first be unloaded and then spend some time in the container yard or, ideally, be immediately transferred to their assigned departing transport unit. In order to avoid bottlenecks and unnecessary delays, the terminal can assign time slots to each incoming transport unit, so that its total service time (the time required to unload it and then load it again and have it ready for departure) is as reduced as possible. The terminal planner knows in advance which containers will arrive on each incoming transport unit, and on which outbound transport unit they must leave the terminal. The service time required for each vessel or train depends on the availability of cranes, and different berth, rail and truck cranes can be assigned to carry out load and unload operations for vessels, trains and trucks. Several cranes may work at the same time on the same vessel or train, but under a single cycle, assumption cranes operating both on trains and vessels can only start loading outgoing containers after all the incoming containers have been previously unloaded. We have assumed that the sequence of loading and unloading operations on vessels is predefined, whereas it needs to be determined for the case of trains, but the formulation is general enough to include different configurations, and the reader may replace “vessels” with “transport units with predefined unload/load sequences” and “trains” with “transport units without predefined unload/load sequences”. Even though the modeling of the problem can have a generic application, we have included several additional assumptions to simplify its mathematical modeling. In the first place, we assume it applies to an intermodal import-export chain, so that containers arriving to the terminal on a train will be transferred to a vessel and vice versa. This does not represent a loss of generality by not considering trucks, since it is always possible to model trucks as trains that carry one single container, and the benefits of establishing slots for trucks in an intermodal terminal have been demonstrated by Zehendner and Feillet [ 29 ]. Furthermore, we have not included the stacking area in the model specifically, but it can be represented by a virtual vessel and train with unlimited capacity, arriving at the beginning of the working day and staying until the end, which absorb all the containers that go to the stacking area and emit all the containers extracted from the stacking area to be loaded on departing transport units. Finally, our concern will not be on individual containers but on container groups, sets of containers that arrive together on the same transport unit and J. Mar. Sci. Eng. 2022,10, 1506 5 of 16 also leave together, and can therefore be treated as an indivisible unit. Given that we do not consider truck or crane displacements in the analysis, which implies ignoring potential conflicts among different elements, we have replaced this assumption with the allocation of one single crane to load or unload the full container group. Again, this assumption could be relaxed by considering only container groups composed of one single container. The objective of the planning process is to schedule the arrivals of vessels and trains and to organize the work sequence of cranes so that vessels and trains finish their loading/unloading operations as early as possible in the terminal. This is why the objective function of our mathematical model depends on the departure times of transport units, which is a typical formulation according to Bierwirth and Meisel [ 30 ], with the arrival times remaining a decision variable. This ultimately results in a reduction of costs, due to the better planning of resources at the terminal, and an increase in the level of service provided by the terminal to transport operators, due to the fast servicing of trains and vessels. The next section describes the mathematical model for this daily planning process, followed by a series of application examples and the metaheuristic methods developed to obtain solutions to real-size problems within acceptable time boundaries. The integrated consideration of several complex problems can be carried out in two different ways. Functional integration [ 31 ] assumes a feedback loop structure or a preprocessing phase, whereby relevant output data from one of the sequential steps is transferred as input to another, and if the results of the different steps are incoherent with one another, the inputs to the first step are modified accordingly and the process is run iteratively until a certain degree of convergence is achieved. A recent example of this approach can be found in Song et al. [ 32 ] for the BAP—QCSP. On the other hand, deep integration merges two subproblems into a partial monolithic problem formulation, which provides a solution for both problems at the same time. This approach was the one chosen by Raa et al. [ 33 ] for an integrated modeling of the BAP and the QCAP. Our work can be viewed as a step in a functional integration sequence, since several parameters of the optimization process developed in this paper are really dependent on the results of that process, such as the amount of time required to load and unload container groups, which is really dependent on the number of cranes assigned to the task, and on the actual scheduling of those cranes, etc., which means that the value for those parameters should be derived from historical data or else be subject to a feedback loop until the results converge. This is due to the complexity of the movement of quay cranes, usually mounted on the same rail and therefore unable to cross one another [ 34 ] and also too large for several of them to work at the same time on the same container group. Finally, the berth allocation step of the process is included in our model by establishing groups of quay cranes, each group corresponding to an individual quay, and forcing vessels to be served by one crane group only. 3. Mathematical Model The formulation of the mathematical model for intermodal terminal planning requires the previous definition of the following sets: Trains: set of trains included in the planning process Vessels: set of vessels included in the planning process TU: set of transport units, with TU =Trains ∪Vessels TC: set of train cranes BC: set of berth cranes BCG: set of berth crane groups C: set of cranes, with C=TC ∪BC CUV: set of container groups that are to be unloaded from vessels CLV: set of container groups that are to be loaded onto vessels CV: set of container groups that are to be moved in vessels, with CV =CUV ∪CLV CUT: set of container groups that are to be unloaded from trains CLT: set of container groups that are to be loaded onto trains CT: set of container groups that are to be moved in trains, with CT =CUT ∪CLT J. Mar. Sci. Eng. 2022,10, 1506 6 of 16 CU: set of container groups that need to be unloaded, with CU =CUT ∪CUV CL: set of container groups that need to be loaded, with CL =CLT ∪CLV The set of variables of the model includes the following: ai= arrival time of transport unit i∈TU di= departure time of transport unit i∈TU uj= starting time for the unloading of container group j∈CU lj= starting time for the loading of container group j∈CL δc kj = 1 if container group k ∈ CU(CL) is unloaded (loaded) immediately before container group j∈CU(CL) on crane c∈C, 0 otherwise µjg = 1 if container group j∈CV is assigned to berth crane group g∈BCG, 0 otherwise Finally, the set of model parameters is as follows: etai= earliest time of arrival of transport unit i ltai=latest time of arrival of transport unit i utj= amount of time required to unload container group j ltj= amount of time required to load container group j Wi= penalty weight for the waiting time of transport unit i M= sufficiently large number αij = binary parameter that indicates whether container group jis to be unloaded from transport unit i βij = binary parameter that indicates whether container group jis to be loaded onto transport unit i ωjk = binary parameter that indicates whether container group jmust be unloaded before container group kfrom the same transport unit γjk = binary parameter that indicates whether container group jmust be loaded before container group konto the same transport unit ζjk = binary parameter that indicates whether container groups jand kare the same when included in two different sets G cg = binary parameter that indicates whether berth crane cbelongs to berth crane group g Then, according to all the above, the objective function for the mathematical model can be formulated as a weighted average of the sum of departure times of the different transport units, as follows: Min ∑ i∈TU Widi(1) The formulation expressed by (1) corresponds to a Capacitated Sum of Completion Times (CSCT) scheduling problem, which adds capacity constraints to the Shortest Processing Time (SPT) problem [35]. This brings together different objectives: n The sum of residence completions for vessels and trains is as small as possible, which implies that vessels and trains will finish their unloading and loading operations and be ready for departure as soon as possible. n Not all the transport units have the same priority, which can be modelled with the W i weights. Typically, for example, the waiting times are more costly for vessels than for trains. n The unloading operations start as soon as possible when the incoming transport unit is ready. Constraints (2) force the arrival of each transport unit to happen within a pre-specified time window: ltai≥ai≥etai∀i∈TU (2) J. Mar. Sci. Eng. 2022,10, 1506 7 of 16 Furthermore, the loading and unloading of container groups can only start after the corresponding transport unit arrives at the terminal. Constraints (3) formulate this condition for unloading operations, and constraints (4) for loading operations. uj≥ai·αij ∀i∈TU,j∈CU (3) lj≥ai·βij ∀i∈TU,j∈CL (4) The following group of constraints establishes that the departure of each transport unit can only take place after all its unloading (constraints (5)) and loading (constraints (6)) operations have been completed: di≥(uj+utj)·αij ∀i∈TU,j∈CU (5) di≥(lj+ltj)·βij ∀i∈TU,j∈CL (6) These next two groups of constraints establish that container groups can only be unloaded (constraints (7)) or loaded (constraints (8)) on vessels after their predecessors have been unloaded or loaded, respectively: uk≥uj·vjk ∀j,k∈CUV (7) lk≥lj·γjk ∀j,k∈CLV (8) Constraints (9) and (10) determine that loading cannot begin in vessels or trains until they have been fully unloaded, and constraints (11) force that container groups can only be loaded on outbound transport units after they have been unloaded from inbound ones. uj≤lk∀j∈CUV,k∈CLV,αij =βik∀i∈Vessels (9) uj≤lk∀j∈CUT,k∈CLT,αij =βik∀i∈Trains (10) (uj+utj)·ξjk ≤lk∀j∈CU,k∈CL (11) All container groups unloaded from or loaded on vessels must be assigned to a group of berth cranes, as expressed by constraints (12): ∑ g∈BCG µjg =1∀j∈CV (12) According to constraints (13), if berth cranes are used, they will be assigned to two virtual tasks: 0 is the virtual initial task for each crane, while 00is the virtual final task. µ0g=µ00g∀g∈BCG (13) All the real unloading and loading tasks will be completed by each group of cranes between those two virtual tasks, under certain restrictions. For instance, constraints (14) establish that if two container groups must be unloaded from the same vessel then both unloading operations must be carried out by the same group of berth cranes. The same condition is expressed by constraints (15) for two loading operations, and by constraints (16) for the case when one container group must be unloaded and the other must be loaded on a given vessel. Constraints (14)–(16) formulate berth allocation in practice, since they establish that vessels must be unloaded and loaded by the same group of berth cranes, and each group of berth cranes corresponds to a given dock. µjg =µkg ∀i∈Vessels,j∈CUV,k∈CUV,αij =αik,g∈BCG (14) µjg =µkg ∀i∈Vessels,j∈CLV,k∈CLV,βij =βik,g∈BCG (15) J. Mar. Sci. Eng. 2022,10, 1506 8 of 16 µjg =µkg ∀i∈Vessels,j∈CUV,k∈CLV,αij =βik,g∈BCG (16) If two container groups are loaded or unloaded simultaneously on a vessel, both operations must be carried out by the same group of berth cranes, as expressed by constraints (17). This includes the fact that the first unloading operation to be completed by a vessel crane must be preceded by the virtual initial task 0, and the last loading operation must be followed by the virtual final task 00. µkg −µjg ≤1−δc kj ∀g∈BCG,c∈BC,j∈CV ∪0, k∈CV ∪00(17) The following groups of constraints, (18) to (21), establish conditions with respect to unloading and loading times; if two container groups are to be processed (loaded or unloaded) consecutively on the same crane, the processing of the second one cannot start until the first one has finished. uk+utk≤uj+M(1−δc kj)∀j,k∈CUV,c∈BC,∀j,k∈CUT,c∈TC (18) uk+utk≤lj+M(1−δc kj)∀j∈CUV,k∈CLV,c∈BC,∀j∈CUT,k∈CLT,c∈TC (19) lk+ltk≤lj+M(1−δc kj)∀j,k∈CLV,c∈BC,∀j,k∈CLT,c∈TC (20) lk+ltk≤uj+M(1−δc kj)∀j∈CLV,k∈CUV,c∈BC,∀j∈CLT,k∈CUT,c∈TC (21) Constraints (22) force that the first task assigned to each train crane is the virtual initial task 0. Then, constraints (23) and (24) establish that each loading/unloading task on a train must have a preceding and a succeeding task, and according to constraint (25), each loading/unloading task carried out on a train crane must have a preceding task and a succeeding task on that same crane. In the formulation of these constraints, we have defined CT0 = CT ∪ 0, and CT0 0 =CT ∪ 0 0 . Constraint (26) closes the cycle by establishing that operations on each train crane must end with the virtual final task 00. ∑ j∈CT δc 0j=1∀c∈TC (22) ∑ c∈TC ∑ k∈CT0,k6=j δc kj =1∀j∈CT (23) ∑ c∈TC ∑ j∈CT00,j6=k δc kj =1∀k∈CT (24) ∑ k∈CT0,k6=j δc kj =∑ p∈CT00,p6=j δc jp ∀j∈CT,c∈TC (25) ∑ j∈CT δc k00=1∀c∈TC (26) Finally, a similar formulation applies to vessel cranes, as expressed by constraints (27) to (31), with CV0 = CV ∪0 and CV00=CV ∪00. ∑ j∈CV δc 0j=1∀c∈VC (27) ∑ k∈CV0,k6=j δc kj =∑ p∈CV00,p6=j δc jp ∀j∈CV,c∈VC (28) ∑ c∈VC ∑ k∈CV0,k6=j δc kj =1∀j∈CV (29) ∑ c∈VC ∑ j∈CV00,j6=k δc kj =1∀k∈CV (30) J. Mar. Sci. Eng. 2022,10, 1506 9 of 16 ∑ j∈CV δc k00=1∀c∈VC (31) 3.1. Model Validation We validated the above model through its application to a series of test problems described in Tables 1and 2. Each table contains a description of the problem data, including the transport units considered (V for vessel and T for train), the container groups that reach and leave the terminal on each transport unit and the time window limits for the arrival of each transport unit to the terminal. Table 3contains the optimal values reached by the objective function and the amount of computational time required to reach it using the Gurobi optimizer on an Intel ® Core ™ I3-4005U, 1.70 GHz computer, depending on the number of berth and rail cranes considered, for both of the previous problem settings. These instances were solved considering all the W i values equal to 1 and all the ut j and lt j values equal to 6. Notably, the required computation time decreases as the number of cranes grows, which corresponds to the fact that finding the optimal scheduling plan is more difficult when the number of resources is scarcer. Table 1. Parameters of scenarios with 2 vessels and 3 trains. Transport Inbound Container Groups Outbound Container Groups Earliest Time of Arrival Latest Time of Arrival V001 ABC IKL 10 40 V002 DEF JGH 20 50 T001 GH EF 5 35 T002 IJ AB 15 45 T003 KL CD 25 55 Table 2. Parameters of scenarios with 2 vessels and 4 trains. Transport Inbound Container Groups Outbound Container Groups Earliest Time of Arrival Latest Time of Arrival V001 ABCD IKLO 10 40 V002 EFGH JMNP 20 50 T001 OP EF 5 35 T002 IJ AB 15 45 T003 KL CD 25 55 T004 MN GH 30 65 Table 3. Computational results given by the model for scenarios with 2 vessels and 3 trains; and 2 vessels and 4 trains. 2 Vessels and 3 Trains 2 Vessels and 4 Trains No of Berth Cranes No of Rail Cranes Optimal Value Computation Time (s) Optimal Value Computation Time (s) 1 1 311 131,066 456 368,475 2 2 220 32,111 307 82,151 3 3 190 8540 260 12,784 4 4 177 101 232 6485 4. Heuristic Solution Procedure Despite the model’s validity for small-scale examples, it soon becomes unsuitable when the total number of container groups Ngrows, since the complexity of the problem is O(N 2 ). When Nincreases, more efficient techniques are required in order to find a suitable solution in a reasonably reduced timeframe. In order to apply heuristic techniques to the problem, we encoded the solutions as data structures containing the following information: J. Mar. Sci. Eng. 2022,10, 1506 16 of 16 8. Kozan, E. Optimising container transfers at multimodal terminals. Math. Comput. Model. 2000,31, 235–243. 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