Currently, air pollution is becoming more critical than ever. Rail transportation is an environmentally friendly solution because of low CO₂ emission, and will be further developed in the near future. In a rail transportation system, the main facility is the rail station. It is the place for trains to be discharged/loaded with outbound/inbound containers by rail cranes (RMGC, RTGC), forklifts and reach stackers. Fig. 1 shows the general layout of a rail station. In this paper, we consider the scheduling of the rail cranes because their operations directly influence the throughput of the rail stations. We not only assigned the job to cranes but also gave the job sequencing for each crane in order to minimize the makespan (the maximum completion time) of cranes. A dual cycle of crane is selected to minimize the number of crane working cycles.

In a working cycle, a crane simultaneously loads and discharges inbound container and outbound container on the trains, respectively. First, the crane moves to the transfer track, picks up the inbound container from the truck and loads it on to the train. Then, the crane moves to the outbound container position, discharges it from the train and loads it on to the truck. The re-handling of container (double handling) happened when an inbound container needs loading on the train but the outbound container at that position has not been unloaded. In that case, the inbound container needs to be loaded to a temporary storage area and be loaded on the train immediately after the outbound container is unloaded.

One difficulty of this problem is re-handling. An inbound container is directly loaded onto a train if the wagon is empty; otherwise, the inbound container needs to be re-handled. In that instance, the inbound container is unloaded from a truck and stacked in a temporary storage area. The inbound container that is stacked in the temporary storage area will be loaded onto the train after the wagon is empty (or the outbound container is discharged). In that case, we say that the outbound needs re-handling. Obviously, the re-handling operations of containers (inbound containers or outbound containers) depend on the point of time that the containers are transferred by cranes. In the simplest case, the re-handling case is caused by the operation of a crane, or may unfortunately be caused by two cranes being operated. A crane loads an inbound container on a wagon but that wagon has yet to be discharged, meaning the inbound container needs to be stored in a temporary storage area. Later, another crane unloads the wagon, and it should load the inbound container onto the wagon.

This paper is organized as follows. Section 2 presents the literature review while Section 3 explains the mathematical model. In Section 4, the Simulated Annealing algorithms are developed to get near optimal solutions. Section 5 presents the performance validation of the algorithms and finally, Section 6 presents the conclusions of this study.

There are numerous studies related to the operations in rail terminals. Research related to the operation problems in a rail station can be classified into three sub-problems of layout design, load planning, and rail crane scheduling. There are some studies related to the layout design of a rail terminal. Kozan [1] determined the crane split and assignment of trains to rail way tracks by a simple heuristic decision rule and a simple dispatching rule. In addition, a simulation was applied based on these heuristic rules to compare different yard layouts. Ballis et al. [2] identified the main design parameters of a railroad transport terminal such as the length and utilization of trans-shipment tracks, the mean stacking height in a storage area, and the type and amount of handling equipment by an analysis tool consisting of three modules comprising of an expert system, a simulation model, and a cost calculation module. Abacoumkin et al. [3] have proposed a similar approach to Ballis et al. [2], where they developed an expert system to define the terminal parameters such as cargo volume, loading unit mixture, equipment, layout characteristic, and operation conditions. Kozan [4] used a simulation model to compare different combinations of handling equipment to determine the most efficient set of handling equipment which is balanced between investigation cost and performance. Benna et al. [5] presented a simulation tool to plan and to design for the infrastructure, capacity, layout, and strategies of railroad container terminals.

There are several papers on the load planning on trains. For instance, Feo et al. [6] optimized the assignments of highway trailer to wagon in a piggyback (trailer on flatcar) system. Bostel et al. [7] were interested in the loading of containers on trains at a rail transfer terminal. Their aim was to minimize the displacement of containers. In contrast to Bostel et al. [7], Corry et al. [8] considered a different terminal system, different objectives and constraints. Their study proposes several techniques for determining an appropriate balance between container handling and weight distribution at a railroad transfer terminal. In Bruns et al. [9], their objective was to assign containers to the wagon of a train to maximize the weight of load units and minimize the travelling cost of load units from their current storage location to the assigned location.

Similarly, there are several papers related to the rail crane scheduling problem. For instance, Alicke [10] delved into logistical issues in container transfer at a theoretical intermodal terminal “Mega Hub” based on constraint satisfaction modeling and heuristics to minimize the maximum lateness of all trains. Boysen et al. [11] investigated the static crane areas in rail—road transshipment yards. In contrast, Boysen et al. [12] focused on the optimization of static crane areas in rail—rail transshipment contexts, using dynamic programming for optimal work area allocation. However, the optimal job sequencing of rail cranes remained unaddressed. Interestingly, none of these studies considered the efficiency-enhancing dual-cycle operation for rail cranes. Jeong et al. [13] demonstrated the application of dual-cycle operations for sequencing rail crane jobs and truck positions, but limited their model to unidirectional crane movement. Pap et al. [14] proposed a branch-and-bound algorithm for optimized dual-cycle job sequencing of a rail cranes, but their work excluded container re-handling scenarios. Guo et al. [15] tackled container loading/unloading scheduling with multiple gantry crane constraints, addressing safety clearance and non-crossing constraints. Their work, however, assumed one-dimensional crane travel and neglected dual-cycle operations or container re-handling. In contrast to these approaches, Nguyen et al. [16] studied rail crane job scheduling specifically in the context of container transfers between trains and trucks, incorporating both re-handling and dual-cycle operations into their model.

Scheduling problems of rail cranes in a rail station are similar to the scheduling problems of quay cranes at sea port container terminals. Many researchers have studied quay crane scheduling problems (QCSP) at port container terminals. In the quay crane scheduling problem, there are several factors which must be considered which are the bays, stacks, groups of containers and containers. However, the majority of studies have focused on the QCSP of container groups or vessel bays. Kim et al. [17] considered the sequence problem of discharging and loading groups of containers of quay cranes in order to minimize the completion time of a ship operation with the interference constraints of cranes. They proposed a branch-and-bound algorithm, and a greedy randomized adaptive search procedure (GRASP) to solve the problem. Moccia et al. [18] revised the formulation in Kim et al. [17], and obtained the optimal solution of the QCSP in small and medium sized instances. A branch-and-cut algorithm was developed to achieve the solutions of larger-sized instances. Moreover, some later studies including those of (Kaveshgar et al. [19]; Lee et al. [20]; Sammarra et al. [21]) tried to revise several issues in the formulation of Kim et al. [17] and applied novel approaches towards problem solving. Ng et al. [22] formulated the quay crane scheduling problem through integer programming. They developed an algorithm to obtain the lower bound of the problem based on the job scheduling problem for parallel machines with sequence independent processing time and LPT (largest processing time) rule. In order to find near optimal solutions, they developed a scheduling heuristic by decomposing the original problem into a number of independent quay crane scheduling sub-problems. The ship was partitioned into zones, with each crane operating within an individual zone. Dynamic programming was used to determine the optimal partition. Meisel [23] considered the QCSP with the inclusion of another aspect, in which a crane at a vessel is only available within certain time windows. A mixed integer programming model as well as a lower bound was developed. A heuristic approach was proposed to minimize the makespan of the cranes. Chen et al. [24] proposed a multi-commodity network flow model with two sets of flow trade-off constraints for multiple cranes and automated guide vehicles. Their constraints were used to solve inter-robot constraints to show the complicated interactions among agents accurately. Yang et al. [25] presented an integrated optimization method to manage the multi-devices integrated scheduling and storage space assignment issue in an energy-efficient manner. A bi-objective model is suggested to minimize the whole operating time and energy consumption, in which the transporting operations of imported and exported intermodal containers were solved simultaneously. Similarly, Li et al. [26] conducted a simulation-based solution for a multiple cranes scheduling issue in a steelmaking shop. Crane scheduling issues for a novel kind of automated container terminal system based on multi-storey frame bridges were explored by Zhen et al. [27]. Schulz et al. [28] depicted the planning situation encountered at a large rail-road terminal in Germany and presented a mixed-integer model capturing the core of the decision-making. They carried out a complicated analysis of issue’s variants and developed a heuristic method in a simulated annealing framework based on these structural insights. Moreover, recent studies on railway container terminal have shown the ongoing interests of various researchers for allocation, planning and scheduling problems [29,30].

In summary, the rail crane scheduling problems in this study are based on the movement of containers. We consider not only the gantry moves, but also the trolley moves. Moreover, the re-handle cases are also considered. In contrast, many studies of QCSP have focused on the operation of quay cranes based on the groups or bays of containers. The movements of containers are usually decided by the crane drivers’ experiences, and hence, reports on QCSP usually only focus on the movement of the gantry. The multi-crane scheduling problem in rail stations is NP-hard, so it is meaningful to propose intelligent optimization algorithm to solve this problem based on parallel simulated annealing.

In this section, the mathematical model is developed.

Notation T

Number of trains (T + 1 is the temporary storage area position; and T + 2 is the transfer track position)

Decision variable

Derived variable

(1) Min[Cmax]

Subject to (2) Ci−Ck≤(1−XiEkk)M∀k∈K,i∈I (3) Ci+Ri×dix,T+2ix,iy−Ck≤(1−XiEkk)M∀k∈K,i∈O (4) Cmax≥Ck+∑i∈ITIik+∑i∈OTOik∀k∈K (5) pi={ty(Tr+2−iy)+Rity(Tr+2−iy)∀i∈ORity+(1−Ri)ty(Tr+2−iy)∀i∈I (6) −MZij≤Ci−Cj+pj<M(1−Zij)∀i,j∈I∪O (7) ∑v=1k∑u∈I∪OXujv−∑v=1k∑u∈I∪OXuiv≤M(Zij+Zji)∀i,j∈I∪O,ix<jx,k∈K (8) ∑k∈K∑i∈I∪OXijk=1∀j∈I∪O (9) ∑i∈I∪OXijk−∑i∈I∪OXjik=0∀k∈K,j∈I∪O (10) ∑i∈I∪OX0kik=1∀k∈K (11) ∑i∈I∪OXiEkk=1∀k∈K (12) Cj−Ci≤RiM+|x(i)−x(j)|M+|y(i)−y(j)|M∀i∈I′,j∈O′ (13) |Ri−Rj|≤|x(i)−x(j)|M+|y(i)−y(j)|M∀i∈I′,j∈O′ (14) Ri=0∀i∈I∖I′ (15) Dij+Ci−Cj≤(1−Xijk)M∀k∈K (16) ∑j∈OXijk(djx,jyix,T+1−djx,jyix,iy−dix,iyix,T+2+dix,T+1ix,T+2)+∑j∈IXijk(djx,T+2ix,T+1−djx,T+2ix,iy−dix,iyix,T+2+dix,T+1ix,T+2)−TIik≤(1−Ri)M∀k∈K,i∈I (17) ∑j∈OXijk(djx,jyix,iy−djx,jyix,T+2+dix,T+2ix,iy)+∑j∈IXijk(djx,T+2ix,iy−djx,T+2ix,T+2+dix,T+2ix,iy)−TOik≤(1−Ri)M∀k∈K,i∈O (18) TIik≥(0−Ri)M∀k∈K,i∈I (19) T0ik≥(0−Ri)M∀k∈K,i∈O (20) dix,T+20kx,0ky+dix,iyix,T+2−Ci<(1−X0ik)M∀k∈K,i∈I (21) dix,iy0kx,0ky+dix,T+2ix,iy−Ci<(1−X0ik)M∀k∈K,i∈O (22) Xijk,Ri,Zij=0 or 1∀k∈K,i,j∈I∪O

The mathematical model is defined in the objective function as Eq. (1). The makespan is the maximum completion time of all cranes, as described in Eq. (4). The completion time of a crane is the completion time of the last container in its container sequence, as in Eqs. (2) and (3). Eq. (5) is the processing time of a container. Eq. (6) defines the variable Zij if container j is handled after the operation of container i is completed, Zij=1; otherwise, Zij=0. Eq. (7) is noninterference constraint. Eq. (8) ensures that a container is assigned to only one crane. Eq. (9) is the flow balance constraint. Eqs. (10) and (11) select the initial and final containers of a crane, respectively. Eqs. (12) to (14) evaluate the re-handling cases. Eq. (12) denotes that if an inbound container i i∈I′ is loaded onto a wagon before the unloading of the outbound container j, Ci≤Cj, j∈O′ and ix=jx and iy=jy, the container i needs to be loaded to the temporary storage area, Ri=1. If the inbound container i and the outbound container have the same position on a train, Ri=Rj (Eq. (13)). If the inbound container i is loaded into temporary storage area, after unloading the outbound container j, the crane should load the inbound container i onto the wagon. Eq. (14) guarantees that the inbound container, which is not occupied by an outbound container, is not loaded into the temporary storage area. Eq. (15) determines the finishing time of a container. Eqs. (16) to (19) define the additional time if a container needs to be re-handled. Eqs. (20) and (21) determine the completion time of the first container in the schedule of a crane. The value of Dij is calculated by Eq. (23). (23) Dij={djx,T+2ix,iy+djx,jyjx,T+2∀i,j∈Idjx,jyix,T+2+djx,T+2jx,jy∀i,j∈Odjx,jyix,iy+djx,T+2jx,jy∀i∈I,j∈Odjx,T+2ix,T+2+djx,jyjx,T+2∀i∈O,j∈I

The meta-heuristic simulated annealing algorithm is used to solve the problem of multiple rail cranes scheduling.

It is very difficult to obtain the optimal solutions from the mathematical model. Fig. 3 shows the computation of CPLEX 12.6 on Windows 10 64 bits, Intel core i7, RAM 24 GB, SSD 512 GB to get the optimal solution from the mathematical model. From Fig. 3, the computation time increases rapidly when the number of containers increases. The computation time is very long because CPLEX used an exact algorithm and the multi-crane scheduling problem in rail stations is NP-hard. Sometimes it is difficult to find the optimum solution with large problem size. For instance, when the number of containers is 10, we cannot achieve the optimal solution within 12 h. With the case of 10 containers and 2 cranes, it takes a very long computational time (more than 16 h) to find the optimal solution.

In this paper, we considered a multi-crane scheduling problem in rail stations. We not only assigned the job tasks to cranes in the dynamic boundary of cranes but also determined the sequence scheduling of each crane. The makespan of cranes was used as an optimization criterion and a Simulated Annealing (SA) and a parallel SA algorithm were proposed to find the solutions. We investigated the performance of algorithms by numerical examples. In general, the parallel SA algorithm outperformed the SA. Furthermore, the algorithms proposed in this paper gave better solutions than other considered algorithms. Moreover, we considered problems with balanced and unbalanced rates of inbound and outbound containers. The problems which had a balance rate of inbound and outbound containers gave smaller makespan, because cranes can perform more dual-cycle operations. When the rate of the outbound container is bigger than the rate of the inbound container, it usually leads to a higher makespan, because of the occurrence of re-handling cases.