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Abstract

In the past, most researchers focused on the storage space allocation problem or container block allocation problem in maritime container terminals, while few studied the container slot allocation problem in rail–water intermodal container terminals. Container slot allocation problem is proposed to reduce relocation operations of containers in railway container yards and improve the efficiency of rail–water intermodal container terminals. In this paper, a novel outbound container slot allocation model is introduced to reduce the rehandling operations, considering stowage plan, containers left from earlier planning periods and container departure time. A novel heuristic algorithm based on the rolling planning horizon approach is developed to solve the proposed problem effectively. Computational experiments are carried out to validate that the proposed model and algorithm are feasible and effective to enhance the storage effect. Meanwhile, some other experiments are conducted to verify that our approach is better than the regular allocation approach, which is a common method in marine and railway container terminals, and container weight is the most important influence factor when storing containers.

Keywords

Intermodal transportation slot allocation problem stowage plan rail–water intermodal container terminals heuristic algorithm

Introduction

Intermodal transportation is defined as the successive use of a variety of modes of transportation, such as road, railway, water and air, without handling the goods during transfers between modes.¹ Nowadays, intermodal transportation has developed rapidly, especially rail–water intermodal transportation.

In 2015, One Belt and One Road has been proposed as an important development initiative by Chinese government. As a result, the number of rail–water intermodal container terminals increases sharply. In rail–water intermodal container terminals, other than quayside area, container yards, inner truck travelling paths and gates, the railway operation area, where many containers transported by train are handled and stored, is also an important part. To save space for rail–water intermodal container terminals, railway container yards are built next to the railway tracks (see Figures 1 and 2), such as Dalian Rail–water Intermodal Container Terminals and Chongqing Rail–water Intermodal Container Terminal. So, rail mounted gantry cranes (RMGCs) are responsible for both handling operations and storage operations. As a result, rational utilization and allocation of railway container yard’s space and handling resources have become dispatcher’s key issues. In addition, because it is difficult to coordinate arrival time or departure time of containers between trains and vessels, outbound and inbound containers transported by train usually need to be stored in railway container yards. Thus, storage space allocation is the vital basis and constraint for other resource utilization in railway container yards.

Figure 1.

An example of the railway container yard in Dalian Rail–water Intermodal Container Terminal.

Figure 2.

A schematic diagram of a railway container yard in a rail–water intermodal container terminal.

As is mentioned by Bazzazi et al.,² storage space allocation problem is a critical scheduling strategy defined as the temporary allocation of inbound/outbound containers to the blocks/slots at each period with the aim of balancing the workload between blocks to minimize the storage/retrieval times of containers. According to the description, storage space allocation problem can be divided into two sub-problems: container block allocation problem and container slot allocation problem. Container block allocation problem deals with assigning the arrival containers to the optimal blocks in the railway container yard, while container slot allocation problem deals with assigning the arrival containers to the specific positions of the block in the railway container yard. The specific positions can also be called slots, which are some small units made up of railway container yards. As a result, the objective of the former is to balance the workload of different blocks, while the latter aims at reducing the number of rehandling operations. In addition, container block allocation usually precedes container slot allocation, while container slot allocation can determine the optimal slots in the railway container yard for each container. Meanwhile, container slot allocation problem is seldom studied in previous studies. Therefore, in this paper, we mainly consider container slot allocation problem of the railway container yard in the rail–water intermodal container terminal.

Because of the structure of vessels, positions of outbound containers in vessels must obey the stowage plan and some other rules during the voyage. This will also lead to storing outbound containers in the railway container yard more complicated. Consequently, only outbound containers are considered in the paper. Meanwhile, when storing outbound containers, weight and arrival port order of containers should be considered according to the stowage plan. Arrival port order means that the order of containers in the same vessel arriving at different destination ports. In addition, departure time of containers and containers left from earlier planning periods should also be considered.

As a result, the main contributions of this paper are (1) the container slot allocation problem is studied to allocate arrival outbound containers to the optimal slots in the railway container yard so that the number of rehandling operations will be reduced; (2) stowage plan (container weight and arrival port order), departure time and containers left from earlier planning periods are considered in the container slot allocation problem and (3) a novel heuristic algorithm (HA) is developed, which introduces the rolling planning horizon.

The remainder of this paper is organized as follows: the relevant literature is presented in next section. The outbound container slot allocation problem in rail–water intermodal container terminals is described and formulated in section ‘Problem description’ and section ‘Problem formulation’, respectively. A novel HA based on the rolling horizon approach is introduced in section ‘Solving algorithm’. Computational results are analysed in section ‘Case study’ and eventually section ‘Conclusion’ covers the conclusion and proposes the future research direction.

Literature review

According to Carlo et al.,³ there are four main decision problems in the storage yard operations: (1) yard design, including material handling equipment (MHE) selection and storage yard layout;^4,5 (2) storage space allocation to containers;^6,7 (3) MHE dispatching and routing to serve the storage and/or retrieval requests^8,9 and (4) optimizing the reshuffling of containers.^10,11

In this paper, we mainly focus on the second one: storage space allocation to containers, which has become one of the hottest issues affecting the efficiency of container terminals. Most of the literatures concentrated on the storage space allocation problem or container block allocation problem in the maritime container terminal to reduce the reshuffling operations and balance the workload. Kim et al.^12,13 discussed the storage space allocation problem in maritime container terminals: the former considered only outbound containers and designed two HAs, while the latter considered inbound and outbound containers together and real-time strategic planning. Kozan and Preston¹⁴ modelled the seaport system to determine the optimal storage strategy and container-handling schedule, which combined a container-transfer model with a container-location model in a planning horizon. A genetic algorithm, a Tabu search and a Tabu search/genetic algorithm hybrid were developed to solve the problem. Lee and colleagues^15,16 developed a yard storage allocation problem in a transhipment hub and proposed a consignment strategy to reduce reshuffling. Two HAs were introduced to solve the problem. Bazzazi et al.² adopted an efficient genetic algorithm to solve an extended storage space allocation problem in a maritime container terminal, considering the type of containers.

Yu and Qi¹⁷ presented the block allocation problem and considered three optimization models under various strategies of storing containers. Then, they used simulation to validate the models and solution approaches. Sharif and Huynh¹⁸ developed a novel method using ant-based control for positioning containers to storage blocks in marine container terminals so that the workload among yard blocks was balanced and the distance travelled by trucks between yard blocks and berths was minimized. Wang et al.¹⁹ studied the storage space allocation problem under the mixed storage mode in railway container terminals and developed a two-stage optimization model. Jiang et al.²⁰ proposed a systematic approach called ‘space allocation given long-term plan (SALP)’ to balance the current and future effects of short-term planning. Two storage strategies were introduced to evaluate the SALP approach. Mao et al.²¹ focused on the storage space allocation problem to enhance the efficiency of the container yard. However, they did not consider the detailed storage allocation constraints.

Moreover, only a few studies were concerned with the container slot allocation problem or rail–water intermodal container terminals. Liu and Yang²² established a two-stage optimization model on multi origin-destination-fare container rail–water intermodal transportation based on revenue management, considering features of freight organization and customer segmentation of container rail–water intermodal transportation. Li et al.²³ developed a container slot allocation model considering the stowage plan in rail–water intermodal container terminals. However, they assumed that bays in container yards were empty before storing the containers. It is unreasonable that the bays are empty; in general, there must be some containers left from earlier planning periods. Meanwhile, they did not consider the departure time of containers. Wang et al.²⁴ studied container slot allocation problem and developed an iterative solution procedure to solve the problem. However, they only considered time constraint and ignored the stowage planning. Wu and colleagues^25,26 concentrated on storage space allocation problem to balance the operation quantity of the container area in rail–water intermodal container terminals. However, they did not consider the specific storage constraints.

Based on all the studies above, most researchers concentrated on the storage space allocation problem in maritime container terminals. Most of them paid attention to the container block allocation problem, which was not to assign arrival containers to the specific slots in the container yard. Although the objective was to reduce the relocation operations or balance the workload of different blocks, they neglected the constraints that affect the container locations in their studies. Meanwhile, only few dealt with the container slot allocation problem. Moreover, specific literature on the storage space allocation problem in railway or rail–water intermodal container terminals is scarce. Some researchers did consider storage principles in the container slot allocation problem; however, they only focused on some of storage principles and few considered weight constraint, arrival port order constraint and departure time constraint together. In addition, most researchers ignored containers left from earlier planning periods in the container yard, which can also affect the storage effect.

Therefore, in this paper, the container slot allocation problem of outbound containers in the rail–water intermodal container terminal is studied, considering not only the stowage plan (the weight and arrival port order of containers), but also containers’ departure time and containers left from earlier planning periods in the railway container yard.

Problem description

Some background

Before formulating a mathematical model, first some concepts and background of the container slot allocation problem will be introduced. As shown in Figures 1 and 2, the railway container yard in a rail–water intermodal container terminal is next to railway tracks, which is made up of some blocks. Figure 3 shows an example of a block. A block is comprised of several parallel bays. Each bay is composed of several stacks and each stack consists of several layers.

Figure 3.

An example of a block.

Like the maritime and railway container terminals, there are two storage modes in rail–water intermodal container terminals as well: the first one is the inbound and outbound containers mixed storage mode, while the second one is the inbound and outbound containers separated storage mode. Although the mixed storage mode can save space for the railway container yard, it will increase the workload of the handling equipment and cut down the efficiency of the whole terminal. Moreover, high quality of storage rules will be required, which increases the difficulty for dispatchers. Meanwhile, in most marine and railway container terminals, inbound and outbound containers are stored separately to improve the efficiency of the storage yard. According to these reasons, we also adopt the inbound and outbound container separated storage mode, and we only consider outbound containers in this paper.

Also, according to Zhang et al.,⁶ a fixed horizon in the immediate future is introduced at each planning epoch, and the plan is executed accordingly up to the next planning epoch; then, a new plan is formulated based on the latest information and this pattern continues. Figure 4 shows an example of the rolling planning horizon. The container slot allocation problem is performed in each period of the planning epoch.

Figure 4.

An example of the rolling planning horizon.

In the maritime container terminal, containers are usually stored based on vessel names and voyage numbers; while in the rail–water intermodal container terminal, containers of different vessels are assumed to be stored in the same bay. So, in a planning period or horizon, not all the containers in the railway container yard can be loaded onto the vessels, and there will be some containers left when the current period or horizon is over. Meanwhile, these containers can also influence the storage of next planning periods or horizons. Therefore, containers left from earlier planning periods also need to be considered in the problem.

Storage principles and overlapping amount

The slot allocation problem is to determine the optimal slot for each container of different vessels so that the rehandling operations of containers will be minimized, the utilization of the railway container yard will be maximized and the efficiency of the rail–water intermodal container terminal will be maximized.

To reduce the relocation operations of outbound containers, the stowage plan also needs to be considered in the slot allocation problem. According to the stowage plan, weight and arrival port orders of containers should also be taken into consideration. Moreover, since containers of different vessels are stored in the same bay, departure time of containers will also be considered. Meanwhile, we assume that when the departure time of different containers is the same, the containers are in the same vessel.

As a result, in this paper, some storage principles should be introduced when containers are allocated into slots of the railway container yard.

According to the stowage plan, the lighter containers usually are stored on the heavier containers in the containership so that the containership can maintain stability; so, in the railway container yard, heavier containers, on the contrary, should be stored on lighter containers to satisfy the stowage plan.

To guarantee the loading order of containers, containers with earlier departure time should be stored above the containers with later departure time in the railway container yard.

When containers have the same departure time, to reduce the relocation operations during navigation, containers with the farther arrival port should be stored above containers with the nearer arrival port in the railway container yard.

Then, we introduce container overlapping, which can be used to evaluate the storage state. Container overlapping means that containers in the same stack do not satisfy one of the storage principles mentioned above: the weight of upper layer containers is lighter than the weight of lower layer containers or the departure time of upper layer containers is later than the departure time of lower layer containers or when upper layer and lower layer containers have the same departure time, the arrival port of upper layer containers is nearer than the arrival port of lower layer containers. When these situations happen, container overlapping generates. Overlapping amount is a numerical value to record the amount of container overlapping.

Also, to describe the attributes of containers concisely, actual weight and departure time can be substituted by weight priority and departure time priority. The heavier the container, the greater the weight priority; the later the container departure time, the greater the departure time priority. In addition, the arrival port order ranks from the near to the distant and numbers from 1. Consequently, the information of the containers consists of the weight priority, departure time priority and arrival port order.

Figure 5 illustrates an example of the overlapping amount. The number outside the parentheses is the number of the container and the numbers inside the parentheses are weight priority, departure time priority and arrival port order. Thus, the overlapping amount in Stack 1 is 1, the overlapping amount in Stack 2 is 2 and the overlapping amount in Stacks 3 and 4 is 1. Stack 1 does not satisfy the third principle, Stack 2 does not satisfy the third storage principle, Stack 3 does not meet the second principle and Stack 4 does not fulfil the first storage principle.

Figure 5.

An example of container overlapping amounts.

Moreover, to avoid that there is no slot available to rehandle containers, it is set that the height of all stacks in the container yard or railway container yard cannot be stacked to the maximum height and enough empty slots must be set aside in the nearest side of each bay for rehandling the containers. Thus, the relationship between layers and empty slots is shown in Figures 6 and 7. As shown in Figures 6 and 7, the grey square box represents the slot stacks a container and the white square box represents an empty slot. So, if the containers in the bottom of Stack 3 need to be picked up, then the containers on it should be removed to the empty slots, which are also called the rehandling slots.

Figure 6.

The rehandling slots of 4-layer storage height.

Figure 7.

The rehandling slots of 5-layer storage height.

So, when there are two-layer containers in a bay, the maximum overlapping amount is the stack in the bay minus 1. As a result, according to the reckoning, when there are $s$ stacks in a bay, the relationship between storage layers and the maximum overlapping amount is shown in Table 1.

Table 1.

The relationship between the maximum overlapping amount and storage layers.

Storage layer	Stack of a bay	Maximum overlapping amount
2	$s$	$s - 1$
3	$s$	$3 s - 4$
4	$s$	$6 s - 9$
5	$s$	$10 s - 16$

Hence, in this paper, we formulate a slot allocation model based on the rolling planning horizons to minimize the total overlapping amounts by considering not only the three storage principles—the weight priority, departure time priority and arrival port order—but containers left from earlier planning periods.

Problem formulation

According to the problem description above, the slot allocation model of outbound containers based on the rolling horizon approach is formulated.

Assumption

The following assumptions are made for formulating the mathematical model:

It is assumed that the size and type of all the containers are the same.

The bay plan of the railway container yard is known in advance; in other words, the positions and attributes of the containers left from earlier planning horizons, such as weight priority and departure time priority, are known before.

The weight priority, departure time priority, arrival port order and arrival time of the new arrival containers are known in advance.

Notations and variables

The notations and variables in the mathematical model are introduced as follows:

Sets.

$M$ : the set of planning periods in a planning horizon, $m \in M$ ;

$I$ : the set of containers that need to be stored in the railway container yard during the period $m$ , $m \in M$ , $i, j \in I$ ;

$B$ : the set of bays in the railway container yard, $b \in B$ ;

$S_{b}$ : the set of stacks in the bay $b$ , $s \in S_{b}$ ;

$T_{b, s}$ : the set of layers in a stack $s$ of the bay $b$ , $t \in T_{b, s}$ ;

Parameters.

$N_{c}^{m}$ : the total number of containers from the arrival trains that need to be stored in the railway container yard during the period $m$ , $m \in M$ ;

$N_{r}^{m}$ : the total overlapping amounts during the period $m$ , $m \in M$ ;

$N_{w}^{m}$ : the overlapping amounts that do not satisfy the weight constraint during the period $m$ , $m \in M$ ;

$N_{e}^{m}$ : the overlapping amounts that do not satisfy the departure time constraint during the period $m$ , $m \in M$ ;

$N_{p}^{m}$ : the overlapping amounts that do not satisfy the arrival port order constraint during the period $m$ , $m \in M$ ;

$N_{w, e}^{m}$ : the overlapping amounts that do not satisfy the weight and departure time constraint simultaneously during the period $m$ , $m \in M$ ;

$N_{w, p}^{m}$ : the overlapping amounts that do not satisfy the weight and arrival port order constraint simultaneously during the period $m$ , $m \in M$ ;

$H_{b, s}$ : the limited height of the stack $s$ in the bay $b$ , $b \in B, s \in S_{b}$ ;

$O_{b, s, t}^{m}$ : the slot status during the period $m$ , if a layer $t$ of the stack $s$ in the bay $b$ is occupied, $O_{b, s, t}^{m} = 1$ ; otherwise, $O_{b, s, t}^{m} = 0$ , $b \in B, s \in S_{b}, t \in T_{b, s}, m \in M$ ;

$W_{i, j}^{m}$ : the status whether containers satisfy the weight constraint during the period $m$ , if the container $i$ and $j$ meet the weight constraint, $W_{i, j}^{m} = 1$ ; otherwise, $W_{i, j}^{m} = 0$ , $i, j \in I, m \in M$ ;

$E_{i, j}^{m}$ : the status whether containers satisfy the departure time constraint during the period $m$ , if the container $i$ and $j$ meet the container departure time constraint, $E_{i, j}^{m} = 1$ ; otherwise, $E_{i, j}^{m} = 0$ , $i, j \in I, m \in M$ ;

$P_{i, j}^{m}$ : the status whether containers with the same departure time satisfy the arrival port order constraint during the period $m$ , if the containers $i$ and $j$ meet the arrival port order constraint, $P_{i, j}^{m} = 1$ ; otherwise, $P_{i, j}^{m} = 0$ , $i, j \in I, m \in M$ ;

Decision variables.

$ϕ_{i, b, s, t}^{m}$ : decision variable, if the container $i$ is stored in a layer $t$ of the stack $s$ in the bay $b$ during the period $m$ , $ϕ_{i, b, s, t}^{m} = 1$ ; otherwise, $ϕ_{i, b, s, t}^{m} = 0$ , $i \in I, b \in B, s \in S_{b}, t \in T_{b, s}, m \in M$ ;

$φ_{i, j, b, s, t_{1}, t_{2}}^{m}$ : decision variable, if the container $i$ and $j$ are stored in the layers $t_{1}$ and $t_{2}$ of the stack $s$ in the bay $b$ during the period $m$ , respectively, $φ_{i, j, b, s, t}^{m} = 1$ ; otherwise, $φ_{i, j, b, s, t}^{m} = 0$ , $i, j \in I, b \in B, s \in S_{b}, t_{1}, t_{2} \in T_{b, s}, m \in M$ ;

$η_{i, j, b, s, t_{1}, t_{2}}^{m, w}$ : decision variable, if the containers $i$ and $j$ are stored in the layers $t_{1}$ and $t_{2}$ of the stack $s$ in the bay $b$ meet the weight constraint during the period $m$ , $η_{i, j, b, s, t_{1}, t_{2}}^{m, w} = 1$ ; otherwise, $η_{i, j, b, s, t_{1}, t_{2}}^{m, w} = 0$ , $i, j \in I, b \in B, s \in S_{b}, t_{1}, t_{2} \in T_{b, s}, m \in M$ ;

$η_{i, j, b, s, t_{1}, t_{2}}^{m, e}$ : decision variable, if the containers $i$ and $j$ are stored in the layers $t_{1}$ and $t_{2}$ of the stack $s$ in the bay $b$ meet the departure time constraint during the period $m$ , $η_{i, j, b, s, t_{1}, t_{2}}^{m, e} = 1$ ; otherwise, $η_{i, j, b, s, t_{1}, t_{2}}^{m, e} = 0$ , $i, j \in I, b \in B, s \in S_{b}, t_{1}, t_{2} \in T_{b, s}, m \in M$ ;

$η_{i, j, b, s, t_{1}, t_{2}}^{m, p}$ : decision variable, if the containers $i$ and $j$ are stored in the layer $t_{1}$ and $t_{2}$ of the stack $s$ in the bay $b$ meet the arrival port order constraint during the period $m$ , $η_{i, j, b, s, t_{1}, t_{2}}^{m, p} = 1$ ; otherwise, $η_{i, j, b, s, t_{1}, t_{2}}^{m, p} = 0$ , $i, j \in I, b \in B, s \in S_{b}, t_{1}, t_{2} \in T_{b, s}, m \in M$ ;

$α_{i, j, b, s, t_{1}, t_{2}}^{m, e}$ : auxiliary variable, the container $i$ and $j$ are stored in the layers $t_{1}$ and $t_{2}$ of the stack $s$ in the bay $b$ during the period $m$ , if they have the same departure time, $α_{i, j, b, s, t_{1}, t_{2}}^{m, e} = 1$ ; otherwise, $α_{i, j, b, s, t_{1}, t_{2}}^{m, e} = 0$ , $i, j \in I, b \in B, s \in S_{b}, t_{1}, t_{2} \in T_{b, s}, m \in M$ .

The slot allocation model of outbound containers

Objective function

Based on the problem description in section ‘Problem description’, the objective function is written as follows

Min N_{r}^{m}

(1)

The objective function of our model is to minimize the overlapping amounts during the period $m$

N_{r}^{m} = N_{w}^{m} + N_{e}^{m} + N_{p}^{m} - N_{w, e}^{m} - N_{w, p}^{m}, \forall b \in B, \forall s \in S_{b}

(2)

\begin{matrix} N_{w}^{m} = \sum_{i, j, s, b, t_{1} < t_{2}} φ_{i, j, s, b, t_{1}, t_{2}}^{m} * (1 - η_{i, j, s, b, t_{1}, t_{2}}^{m, w}), \\ \forall i, j \in I, \forall b \in B, \forall s \in S_{b}, \forall t_{1}, t_{2} \in T_{b, s}, \forall m \in M \end{matrix}

(3)

\begin{matrix} N_{e}^{m} = \sum_{i, j, s, b, t_{1} < t_{2}} φ_{i, j, s, b, t_{1}, t_{2}}^{m} * (1 - η_{i, j, s, b, t_{1}, t_{2}}^{m, e}), \\ \forall i, j \in I, \forall b \in B, \forall s \in S_{b}, \forall t_{1}, t_{2} \in T_{b, s}, \forall m \in M \end{matrix}

(4)

\begin{matrix} N_{p}^{m} = \sum_{i, j, s, b, t_{1} < t_{2}} φ_{i, j, s, b, t_{1}, t_{2}}^{m} * (1 - η_{i, j, s, b, t_{1}, t_{2}}^{m, p}) * α_{i, j, s, b, t_{1}, t_{2}}^{m, e}, \\ \forall i, j \in I, \forall b \in B, \forall s \in S_{b}, \forall t_{1}, t_{2} \in T_{b, s}, \forall m \in M \end{matrix}

(5)

\begin{matrix} N_{w, e}^{m} = \sum_{i, j, s, b, t_{1} < t_{2}} φ_{i, j, s, b, t_{1}, t_{2}}^{m} * (1 - η_{i, j, s, b, t_{1}, t_{2}}^{m, w}) \\ * (1 - η_{i, j, s, b, t_{1}, t_{2}}^{m, e}), \\ \forall i, j \in I, \forall b \in B, \forall s \in S_{b}, \forall t_{1}, t_{2} \in T_{b, s}, \forall m \in M \end{matrix}

(6)

\begin{matrix} N_{w, p}^{m} = \sum_{i, j, s, b, t_{1} < t_{2}} φ_{i, j, s, b, t_{1}, t_{2}}^{m} * (1 - η_{i, j, s, b, t_{1}, t_{2}}^{m, w}) \\ * (1 - η_{i, j, s, b, t_{1}, t_{2}}^{m, p}) * α_{i, j, s, b, t_{1}, t_{2}}^{m, e}, \\ \forall i, j \in I, \forall b \in B, \forall s \in S_{b}, \forall t_{1}, t_{2} \in T_{b, s}, \forall m \in M \end{matrix}

(7)

Equation (2) means that the overlapping amounts consist of container overlapping caused by storage constraints violation. Equation (3) indicates the calculation of the overlapping amounts violating the weight constraint during the period $m$ . Similarly, Equations (4) and (5) describe the overlapping amounts that do not meet the departure time constraint and the arrival port order constraint during the period $m$ , respectively. Equation (6) shows the calculation of the overlapping amounts violating both the weight constraint and departure time constraint. Similarly, Equation (7) calculates the overlapping amounts that do not meet the weight constraint and arrival port order constraint at the same time.

Constraints

The constraints of the model are proposed as follows to ensure the practical feasibility of the solution.

1. Allocation operation constraints.

A slot in the railway container yard can be occupied by at most one container

\sum_{i} ϕ_{i, b, s, t}^{m} \leq 1 - O_{b, s, t}^{m}, \forall b \in B, \forall s \in S_{b}, \forall t \in T_{b, s}, \forall m \in M

(8)

Meanwhile, one container can occupy only one slot in the railway container yard

\sum_{b, s, t} ϕ_{i, b, s, t}^{m} \leq 1, \forall i \in I, \forall m \in M

(9)

2. Workload constraints.

The number of containers to be stored in the railway container yard equals to the number of containers from the arrival trains during the period $m$

\sum_{i, b, s, t} ϕ_{i, b, s, t}^{m} = N_{c}^{m}, \forall m \in M

(10)

3. Allocation position constraints.

Containers in the railway container yard cannot be assigned on the empty slots

\sum_{i} ϕ_{i, b, s, t}^{m} + O_{b, s, t}^{m} \geq \sum_{i} ϕ_{i, b, s, (t + 1)}^{m}, \forall b \in B, \forall s \in S_{b}, \forall t \in T_{b, s}, \forall m \in M

(11)

The slot must be empty before container $i$ is stored in it

ϕ_{i, b, s, t}^{m} + O_{b, s, t}^{m} = 1, \forall i \in I, \forall b \in B, \forall s \in S_{b}, \forall t \in T_{b, s}, \forall m \in M

(12)

The storage height of a stack $s$ should not exceed the limited height of the stack $s$

\sum_{i, t} ϕ_{i, b, s, t}^{m} + \sum_{t} O_{b, s, t}^{m} \leq H_{b, s}, \forall b \in B, \forall s \in S_{b}, \forall m \in M

(13)

In addition, the relationship between decision variables $ϕ_{i, b, s, t}^{m}$ and $φ_{i, j, b, s, t_{1}, t_{2}}^{m}$ is shown as follows

ϕ_{i, b, s, t_{1}}^{m} + ϕ_{j, b, s, t_{2}}^{m} - φ_{i, j, b, s, t_{1}, t_{2}}^{m} \leq 1, \forall i, j \in I, \forall b \in B, \forall s \in S_{b}, \forall t_{1}, t_{2} \in T_{b, s}, \forall m \in M

(14)

4. Allocation preference constraints.

The relationship among $φ_{i, j, b, s, t_{1}, t_{2}}^{m}$ , $η_{i, j, b, s, t_{1}, t_{2}}^{m, w}$ and $W_{i, j}^{m}$ : if the containers $i$ and $j$ are stored in the same stack and the weight of the container $i$ is lighter than the weight of the container $j$ , $η_{i, j, b, s, t_{1}, t_{2}}^{m, w}$ equals to 1; otherwise, it equals to 0

\begin{matrix} W_{i, j}^{m} + φ_{i, j, b, s, t_{1}, t_{2}}^{m} - η_{i, j, b, s, t_{1}, t_{2}}^{m, w} \leq 1, \\ \forall i, j \in I, \forall b \in B, \forall s \in S_{b}, \forall t_{1} < t_{2} \in T_{b, s}, \forall m \in M \end{matrix}

(15)

\begin{matrix} W_{i, j}^{m} * φ_{i, j, b, s, t_{1}, t_{2}}^{m} - η_{i, j, b, s, t_{1}, t_{2}}^{m, w} \geq 0, \\ \forall i, j \in I, \forall b \in B, \forall s \in S_{b}, \forall t_{1} < t_{2} \in T_{b, s}, \forall m \in M \end{matrix}

(16)

The relationship among $φ_{i, j, b, s, t_{1}, t_{2}}^{m}$ , $η_{i, j, b, s, t_{1}, t_{2}}^{m, e}$ and $E_{i, j}^{m}$ : if the containers $i$ and $j$ are stored in the same stack and the container $i$ departs later than the container $j$ , $η_{i, j, b, s, t_{1}, t_{2}}^{m, e}$ equals to 1; otherwise, it equals to 0

\begin{matrix} E_{i, j}^{m} + φ_{i, j, b, s, t_{1}, t_{2}}^{m} - η_{i, j, b, s, t_{1}, t_{2}}^{m, e} \leq 1, \\ \forall i, j \in I, \forall b \in B, \forall s \in S_{b}, \forall t_{1} < t_{2} \in T_{b, s}, \forall m \in M \end{matrix}

(17)

\begin{matrix} E_{i, j}^{m} * φ_{i, j, b, s, t_{1}, t_{2}}^{m} - η_{i, j, b, s, t_{1}, t_{2}}^{m, e} \geq 0, \\ \forall i, j \in I, \forall b \in B, \forall s \in S_{b}, \forall t_{1} < t_{2} \in T_{b, s}, \forall m \in M \end{matrix}

(18)

The relationship among $φ_{i, j, b, s, t_{1}, t_{2}}^{m}$ , $η_{i, j, b, s, t_{1}, t_{2}}^{m, p}$ , $α_{i, j, b, s, t_{1}, t_{2}}^{m, e}$ and $P_{i, j}^{m}$ : if the containers $i$ and $j$ are stored in the same stack and the arrival port of the container $i$ is earlier than that of the container $j$ , $η_{i, j, b, s, t_{1}, t_{2}}^{m, p}$ equals to 1; otherwise, it equals to 0

\begin{matrix} P_{i, j}^{m} + φ_{i, j, b, s, t_{1}, t_{2}}^{m} - η_{i, j, b, s, t_{1}, t_{2}}^{m, p} * α_{i, j, b, s, t_{1}, t_{2}}^{m, e} \leq 1, \\ \forall i, j \in I, \forall b \in B, \forall s \in S_{b}, \forall t_{1} < t_{2} \in T_{b, s}, \forall m \in M \end{matrix}

(19)

\begin{matrix} P_{i, j}^{m} * φ_{i, j, b, s, t_{1}, t_{2}}^{m} - η_{i, j, b, s, t_{1}, t_{2}}^{m, p} * α_{i, j, b, s, t_{1}, t_{2}}^{m, e} \geq 0, \\ \forall i, j \in I, \forall b \in B, \forall s \in S_{b}, \forall t_{1} < t_{2} \in T_{b, s}, \forall m \in M \end{matrix}

(20)

5. Decision variable constraints.

The decision variable constraint is shown as follows

\begin{matrix} ϕ_{i, b, s, t}^{m}, φ_{i, j, b, s, t_{1}, t_{2}}^{m}, η_{i, j, b, s, t_{1}, t_{2}}^{m, w}, η_{i, j, b, s, t_{1}, t_{2}}^{m, e}, η_{i, j, b, s, t_{1}, t_{2}}^{m, p}, \\ α_{i, j, b, s, t_{1}, t_{2}}^{m, e} \in [0, 1], \\ \forall i, j \in I, \forall b \in B, \forall s \in S_{b}, \forall t_{1} < t_{2} \in T_{b, s}, \forall m \in M \end{matrix}

(21)

Solving algorithm

Since the rolling planning horizon is considered in the paper and we consider the characteristics of the proposed model, a novel HA is developed to obtain a close optimal solution effectively.

During the storage process, the arrival port order is considered when the departure time of containers is the same, so the departure time is the prerequisite; however, there is no compulsory order requirement for the weight priority and departure time priority. Thus, to describe the algorithm in more detail, an example is introduced: here, we take the first principle as the most important impact factor during the storage process and the flow chart of the HA is shown in Figure 8.

Figure 8.

Flowchart of the heuristic algorithm based on the rolling horizon approach.

The procedure of HA

The notations used in the description of the procedure and the detailed description of the procedure are as follows:

$I_{w}$ : the set of weight priority in an ascending order: $I_{w} = {i_{w min}, \dots, i_{w}, \dots, i_{w \max}}$ ; $i_{w}$ means the container with the specified weight priority.

$E$ : the set of departure time priority with the same weight priority in an ascending order, $E = {i_{w, min}, \dots, i_{w, e}, \dots, i_{w, e \max}}$ ; $i_{w, e}$ means the container with the specified departure time priority.

$P$ : the set of destination ports with the same departure time priority in an ascending order, $P = {i_{w, e, min}, \dots, i_{w, e, p}, \dots, i_{w, e, p \max}}$ ; $i_{w, e, p}$ means the container with the specified destination port.

$T$ : the set of tiers in a stack $s$ of the bay $b$ , $T = {t_{min}, \dots, t, \dots, t_{\max}}$ , $t_{\max} = H_{b, s}$ .

$A_{0}$ : the initial storage matrix.

$n_{c}^{m}$ : the number of the container during the period $m$ .

$A_{n_{c}^{m}}^{m}$ : the storage matrix after allocating the container during the period $m$ .

$N_{r}^{m}$ : the overlapping amount after allocating this container.

$N_{r'}^{m}$ : the overlapping amount before allocating this container.

Step 1. Input the planning horizon $M$ , initialize the planning period $m$ and set $m = 1$ .

Step 2. Input the initial storage matrix $A_{0}$ , the number of containers $n_{c}^{m} = 0$ and the overlapping amount $N_{r}^{m} = 0$ .

Step 3. According to the first storage principle, traverse the weight priority of all the containers from the containers left from earlier planning horizons and the new arrival containers during the period $m$ and sort them: $I_{w}$ .

Step 4. Choose the lightest container $i_{w}$ from the set $I_{w}$ . If there is only one container $i_{w}$ and it has already been in the bay, then go to Step 13.

Step 5. If there is more than one container $i_{w}$ , then go to Step 11; otherwise, locate the container in the assigned bay.

Step 6. Sort the layers in the selected bay: $T$ . Select the layer $t$ from the bottom to the top.

Step 7. Check whether there is an empty slot in the layer $t$ , if there is no empty slot, then go to Step 10.

Step 8. If there is only one empty slot, then allocate this container to this slot, update the storage matrix $A_{n_{c}^{m}}^{m}$ and go to Step 14; otherwise, there is over one slot.

Step 9. Select the best slot from these empty slots. Compare the overlapping amount before allocating this container $N_{r'}^{m}$ and the overlapping amount after allocating this container $N_{r}^{m}$ ; if $N_{r'}^{m} \geq N_{r}^{m}$ , then allocate the container in the slot. Update the storage matrix $A_{n_{c}^{m}}^{m}$ and go to Step 15. Otherwise, $N_{r'}^{m} < N_{r}^{m}$ , then allocate the container into the slot whose weight priority difference between the container and its lower layer containers is the minimum, and update the storage matrix $A_{n_{c}^{m}}^{m}$ and go to Step 13.

Step 10. Update the layer $t = t + 1$ of the bay, if the layer $t < H_{b, s}$ , then go to Step 7; otherwise, select another bay and go to Step 6.

Step 11. Traverse the departure time priority with the same weight priority and sort them: $E$ ; then, choose the container with the latest departure time $i_{w, e}$ ; if there is only one container $i_{w, e}$ , then go to Step 6.

Step 12. Traverse the arrival port order with the same departure time priority and sort them: $P$ ; then, allocate them in the ascending order and go to Step 6.

Step 13. Update the total number of containers $n_{c}^{m} = n_{c}^{m} + 1$ , if $n_{c}^{m} \leq N_{c}^{m} + \sum_{b, s, t} O_{b, s, t}^{m}$ , then go to Step 3; otherwise, go to Step 15.

Step 14. If the new container and its lower layer containers satisfy the storage principles, then go to Step 13; otherwise, update the overlapping amount $N_{r}^{m}$ and go to Step 13.

Step 15. Update the planning period $m = m + 1$ , if $m \leq M$ and then go to Step 2.

Step 16. Storage task ends. Output the overlapping amount $N_{r}^{m}$ .

The pseudo code of the HA

The pseudo code of HA is shown in Appendix 1.

Case study

In this section, three groups of computational experiments are implemented to verify the proposed model and algorithm. The data we used in all these computational experiments came from the survey data of one of China famous railway container terminals. All the computational experiments are conducted based on a personal computer with Intel Core i5-2520M at 2.50 GHz processors and 4GB RAM.

Although in this paper, the scheduling of RMGCs is not considered, they usually travel on the same track and cannot cross each other. Therefore, to guarantee the safety and feasibility of storage operations, we regulate that RMGCs store and handle containers in a fixed region of the railway container yard so that RMGCs can store containers simultaneously without affecting each other. Consequently, each RMGC is responsible for the assigned bays; for example, if there are 3 RMGCs in the railway container yard, then each RMGC takes charge of about 20 bays.

In all the following experiments, we set 3 days as a planning horizon, 1 day as a planning epoch and 6 h as a planning period. There are 4 planning periods in one planning epoch and 12 planning periods in one planning horizon. Meanwhile each bay has 4 stacks and 4 layers. As is mentioned in section ‘Storage principles and overlapping amount’, when there are 4-layer containers in the bays, it is essential for the bays to have 3 empty slots. So, one bay can only store 13 containers.

First, to verify the feasibility of the model and algorithm, a medium-sized sample of 3 days with 72 containers left from earlier planning horizons and 180 new arrival containers is carried out. Meanwhile, the 180 new arrival containers are also divided into four groups. The first group of 50 containers arrives in the period $m = 1$ ; the second group of 40 containers arrives in the period $m = 4$ ; the third group of 45 containers arrives in the period $m = 7$ and the fourth group of 45 containers arrives in the period $m = 10$ . The detailed information of containers left from earlier planning periods and the new arrival containers are shown in Appendix 2.

Here, we introduce the overlapping rate $χ^{m}$ , which is used to describe the storage effect. The smaller the ratio value, the better the storage effect. The calculation of the overlapping rate $χ^{m}$ is as follows

χ^{m} = \frac{N_{r}^{m}}{N_{c}^{m} + \sum_{b, s, t} O_{b, s, t}^{m}} * 100 %

(22)

According to the proposed HA, the overlapping amounts and overlapping rates during one planning horizon are shown in Table 2.

Table 2.

The overlapping amounts and the overlapping rates in each planning period.

Planning period	Overlapping amounts	The numbers of containers	Overlapping rates (%)
1	21	122	17.2
2	21	119	17.6
3	21	111	18.9
4	30	129	23.3
5	23	96	24.0
6	18	74	24.3
7	15	99	15.2
8	9	77	11.7
9	4	57	7.0
10	4	75	5.3
11	3	58	5.2
12	3	40	7.5

As observed in Table 2, the overlapping rates are all lower than 25%, and the average overlapping rate is 14.8%. So, HA can obtain good storage effect.

Take the planning period $m = 7$ as an example, the detailed overlapping amounts in each bay are shown in Table 3. Then take RMGC 3 as an example. In planning period $m = 7$ , 15 containers are assigned to the slots of 3 bays by RMGC 3, and meanwhile, there are 6 containers left from the earlier planning periods in these 3 bays. The container storage states of these three bays are shown in Figures 9 –11.

Table 3.

The overlapping amounts and overlapping rates in each bay during the planning period $m = 7$ .

Bay	Overlapping amounts	The number of containers	Overlapping rates (%)
Bay 3	0	6	0.0
Bay 4	0	2	0.0
Bay 8	0	3	0.0
Bay 10	0	7	0.0
Bay 14	2	8	25.0
Bay 18	0	4	0.0
Bay 23	3	9	33.3
Bay 24	3	5	60.0
Bay 28	1	7	14.3
Bay 30	3	10	30.0
Bay 34	0	3	0.0
Bay 38	1	6	16.7
Bay 43	0	7	0.0
Bay 44	0	2	0.0
Bay 48	0	6	00.0
Bay 50	1	2	50.0
Bay 54	1	8	12.5
Bay 58	0	4	0.0

Figure 9.

The container storage state in bay 43 (the overlapping amount is 0).

Figure 10.

The container storage state in bay 48 (the overlapping amount is 0).

Figure 11.

The container storage state in bay 54 (the overlapping amount is 1).

To illustrate the performance of the proposed model and algorithm, an experiment is implemented between the proposed algorithm and the regular allocation approach (RAA). RAA is to allocate the containers to the empty slots of the bay one by one according to containers’ arrival time and the arrival order of each container.

In this experiment, another medium-sized sample of 4 days with 81 containers left from earlier planning horizons and 279 new arrival containers is introduced. Meanwhile, the 279 new arrival containers are also divided into 6 groups. The first group of 50 containers arrives in the period $m = 1$ , the second group of 45 containers arrives in the period $m = 4$ , the third group of 40 containers arrives in the period $m = 8$ , the fourth group of 48 containers arrives in the period $m = 11$ , the fifth group of 51 containers arrives in the period $m = 13$ and the sixth group of 45 containers arrives in the period $m = 16$ .

We introduce the value $GA P_{1}$ to evaluate the difference of the overlapping amount between RAA and HA, which implies the improved value of the overlapping amount. The calculation of the value $GA P_{1}$ and the notations used in the calculation are as follows

G A P_{1} = \frac{O V_{R A A} - O V_{H A}}{O V_{R A A}} * 100 %

(23)

$O V_{RAA}$ is the total overlapping amount in RAA, $O V_{HA}$ is the total overlapping amount in HA and $GA P_{1}$ is the difference between $O V_{RAA}$ and $O V_{HA}$ .

The results of the experiment are shown in Table 4 and Figure 12.

Table 4.

The results of the experiment between HA and RAA.

Planning period	OV_HA	OV_RAA	GAP ₁ (%)	The number of containers
1	22	29	24.1	114
2	19	29	34.5	98
3	13	23	43.5	81
4	24	44	45.5	111
5	17	38	55.3	92
6	23	47	51.1	114
7	22	46	52.2	106
8	18	41	56.1	98
9	28	59	52.5	131
10	22	58	62.1	121
11	21	47	55.3	107
12	40	72	44.4	141
13	36	75	52.0	125
14	29	45	35.6	100
15	16	33	51.5	79
16	18	39	53.8	111

HA: heuristic algorithm; RAA: regular allocation approach.

Figure 12.

The overlapping amount between HA and RAA.

As shown in Figure 12 and Table 4, when there are some new arrival containers during the planning period, both $O V_{HA}$ and $O V_{RAA}$ increase; however, $O V_{RAA}$ increases more rapidly than $O V_{HA}$ . Meanwhile, when there are only departure containers during the planning period, although $O V_{RAA}$ decreases more rapidly than $O V_{HA}$ , all $O V_{RAA}$ are still much more than $O V_{HA}$ . All $GA P_{1}$ are more than 20% and the average $GA P_{1}$ is 48.1%. In addition, the computational time of HA is much faster than the computational time of RAA.

As a result, the proposed algorithm performs better in reducing the overlapping amounts when storing containers.

In section ‘The procedure of HA’, we only take the weight priority as an example to describe the proposed algorithm. Therefore, to determine which storage principle is the most important factor, another experiment between the weight priority and departure time priority is implemented based on the data from the second experiment. To evaluate the performance of the comparative experiments, the value $GA P_{2}$ is introduced, which means the difference in the total overlapping amounts between weight priority and departure time priority. The calculation of the value $GA P_{2}$ and the notations used in the calculation are shown as follows

G A P_{2} = \frac{O V_{e} - O V_{w}}{O V_{e}} * 100 %

(24)

$O V_{w}$ is the total overlapping amount in the experiment with the weight priority as the first factor, $O V_{e}$ is the total overlapping amount in the experiment with the departure time priority as the first factor and $GA P_{2}$ is the difference between $O V_{w}$ and $O V_{e}$ . The results of the experiment are shown in Table 5 and Figure 13.

Table 5.

The results of the experiment between HA and RAA.

Planning period	$O V_{w}$	$O V_{e}$	$GA P_{2}$ (%)	The total number of containers
1	22	22	0.0	114
2	19	19	0.0	98
3	13	14	7.1	81
4	24	32	25.0	111
5	17	21	19.0	92
6	23	28	17.8	114
7	22	28	21.4	106
8	18	25	28.0	98
9	28	34	17.6	131
10	22	28	21.4	121
11	21	23	8.7	107
12	40	44	9.0	141
13	36	37	2.7	125
14	29	30	3.3	100
15	16	19	15.8	79
16	18	17	−5.8	111

HA: heuristic algorithm; RAA: regular allocation approach.

Figure 13.

The overlapping amount between the weight priority and departure time priority.

It can be obviously found from Table 5 and Figure 13 that $O V_{w}$ is less than $O V_{e}$ in most situations. Meanwhile, the average $GA P_{2}$ is 11.9%. Hence, when storing containers, the weight priority should be fulfilled first; then, the departure time priority should be satisfied; finally, the arrival port order needs to be met.

Conclusion

In this paper, we focus on the outbound container slot allocation problem in rail–water intermodal container terminals considering the stowage plan, departure time priority and containers left from earlier planning horizons. Then, a mathematical model based on the rolling horizon planning is proposed to minimize the total overlapping amounts. Since the rolling planning horizon is also considered, a novel HA based on the rolling horizon approach is introduced to solve the proposed problem effectively.

Three groups of computational experiments are conducted. The results of the first experiment showed that overlapping rates are all lower than 25.0%. Meanwhile, the results of the second experiment that compare the proposed method with the RAA verify that the proposed method performs better in both storage states and computational time. In addition, we also determine that the weight priority is the most important factor that influences container storage state by the third experiments.

In future, we will consider inbound and outbound containers together. Another research direction is to consider some uncertain factors, such as uncertain arrival time of containers, and combine railway containers with road containers together in the same model.

Footnotes

Appendix 1

Table 6.

The pseudo code of the heuristic algorithm based on the rolling horizon approach.

1: input the planning horizon M and the initial storage matrix A 0

2: set m ← 1 , n c m ← 1 and N r m ← 0

3: for m ← 1 t o M do

4: for n c m ← 1 t o N c m + ∑ b , s , t O b , s , t m do

5: traverse the weight priority of containers and sort them I w ← { i w max , . . . , i 2 , i 1 }

6: for i w ← i w min t o i w max do

7: if i w is the only one then

8: if i w is in the bay then

9: continue

10: else

11: sort layer t : T ← { 1 , 2 , . . . , t max } in the selected bay and set t max ← H b , s

12: for t ← 1 t o t max do

13: if there is an empty slot in layer t then

14: allocate container i w

15: update the storage matrix A n c m m

16: if | i w , e − i w ′ , e ′ | ≠ 0 (container i w ′ , e ′ is the container in lower layer) then

17: if i w ′ , e ′ < i w , e then

18: update the overlapping amount N r m

19: end if

20: else

21: if i w ′ , e ′ , p ′ > i w , e , p (container i w ′ , e ′ , p ′ is the container in the lower layer) then

22: update the overlapping amount N r m

23: end if

24: end if

25: else there is no empty slot in layer t then

26: continue

27: else if there are more than one slots in layer t then

28: if N r m ≤ N r ′ m ( N r ′ m is the former overlapping amount) then

29: allocate container i w in the slot

30: update the storage matrix A n c m m

31: else

32: select the slot whose ( i w ′ − i w ) is minimum ( i w ′ is the container in the lower layer)

33: end if

34: end if

35: end for

36: end if

37: else

38: traverse the departure time priority of containers with the same weight and sort them

E ← { i w , 1 , i w , 2 , . . . , i w , e max }

39: for i w , e ← i w , e min t o i w , e max do

40: if i w , e is the only one then

41: allocate i w , e in the slot

42: update the storage matrix A n c m m and the overlapping amount N r m

43: else

44: allocate the container in an ascending order of P ← { i w , e , 1 , i w , e , 2 , . . . , i w , e , p max }

45: update the storage matrix A n c m m and the overlapping amount N r m

46: end if

47: end for

48: end if

49: end for

50: end for

51: end for

Appendix 2

The detailed information of containers left from earlier planning periods and new arrival containers used in the first experiment is shown in Tables 7 and 8, respectively.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

This work was supported by the Fundamental Research Funds for the Central Universities (grant number 2018YJS080), the Major Research Plan of the National Natural Science Foundation of China (grant number 71390332) and the National Key R&D Programme of China (grant number 2018YFB1201403).

ORCID iD

Yimei Chang

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The outbound container slot allocation based on the stowage plan in rail–water intermodal container terminals

Abstract

Keywords

Introduction

Literature review

Problem description

Some background

Storage principles and overlapping amount

Problem formulation

Assumption

Notations and variables

The slot allocation model of outbound containers

Objective function

Constraints

Solving algorithm

The procedure of HA

The pseudo code of the HA

Case study

Conclusion

Footnotes

Appendix 1

Appendix 2

Declaration of conflicting interests

Funding

ORCID iD

References