Integrated Containership Stowage Planning: A Methodology for Coordinating Containership Stowage Plan and Terminal Yard Operations

Abstract

This study aims at achieving the optimized stowage plan based on the containers’ distribution in the yard and space structure of the ship by integrating the containership stowage problem with the block relocation and loading problem. The containership’s physical structure contributes to a variety of stowage plans, and the containers’ distribution in the yard restrains the container loading sequence. The integrated containership stowage model is established, considering irregular format ship configuration, the weight of each individual container and the stability constraints by limiting the range of related indicators to strengthen real-world scalability. Commercial managers can solve small-scale instances to reduce handling moves in the terminal yard by using the mathematical model, but they cannot solve large-scale instances in the real world. Thus, this study presents a heuristic algorithm to coordinate containership stowage and terminal yard operations. Numerical experiments compare the optimal solutions using the heuristic algorithm to that by CPLEX and GUROBI in small-scale instances, and the respective results are displayed in two stages using the heuristic algorithm. Computational results show that the proposed algorithm can provide the industry with decision support by dramatically and quickly reducing the relocations in the yard and the shifts in the containership, helping the container terminal keep a sustainable development process.

1. Introduction and Literature Review

UNCTAD [1] reported that the world container terminal throughput in 2020 (798.87 million TEUs) slightly decreased by 1.05% compared to the throughput in 2019 (807.33 million TEUs) and has dramatically increased by 47.46% compared to that in 2010 (541.76 million TEUs). In recent years, container terminals have become busy and are faced with high throughput. Thus, one of container terminals’ goals is to improve operational efficiency, considering each terminal yard’s and containership’s handling operations simultaneously.

Relocations in container yards, causing unproductive operations, have an important effect on container terminals’ operational efficiency. The block (container) relocation problem (BRP, CRP) was first proposed by Kim and Hong [2], and it only involves the yard stacks and yard tiers of a yard bay, which is defined as a two-dimensional problem. Thus, Caserta et al. [3] considered the parameters of width and height, which are the two dimensions of the established model. Jin et al. [4,5] focused on reducing relocations by a greedy heuristic integrated with the look-ahead rule to create the crane’s optimal operation plan. Caserta et al. [6] presented the BRP-II model, which was later corrected by Zehendner et al. [7], and designed a simple heuristic integrated with some relocation rules based on realistic assumptions. Moreover, Forster and Bortfeldt [8] proposed a tree search algorithm for a similar BRP that assumed the containers were classified into groups. Wang et al. [9], Exposito-Izquierdo et al. [10] and Lee and Hsu [11] proposed the container pre-marshalling problem (CPMP) to avoid further relocations by predetermining the arrangement of all the containers in the storage space. Petering and Hussein [12] integrated the pre-marshalling rules in CPMP to the BRP and obtained superior solutions by a new proposed look-ahead algorithm. In addition, other variants of BRP appeared, such as dynamic BRP (DBRP) [13], stochastic BRP (SBRP) [14], BRP with time window (BRPTW) [15] and so on. To sum up, all of the BRP and CPMP, etc., aim to optimize the number of crane moves when moving containers from the storage space.

The containers’ distribution in the ship is worthy of attention. Making an optimal stowage plan is an effective way to improve container terminal efficiency by minimizing the number of shifts when unloading containers in the ship. The appropriate stowage plan is helpful to decrease the turn round time of a ship in a port of call. For a single ship bay, the 2D containership stowage problem was performed, not considering the seaworthy constraints. Avriel et al. [16] obtained a stowage plan using a new proposed suspensory heuristic embedded with a dynamic slot assignment rule, which was improved by Ding and Chou [17]. However, the 3D containership stowage problem involves the containers’ distribution of each ship bay in each port of the shipping route, considering the consequences of the stowage plan at the loading port on decisions at subsequent ports. Wilson and Roach [18] proposed a two-phase approach to achieve the optimal stowage plan automatically. Most research is focused on the master bay planning of the two-phase approach: Sciomachen and Tanfani [19] developed integer programming (IP) models, which can be used to optimize small-scale instances; Imai et al. [20] established a multi-objective function model for estimation of container rehandles and ship stability; and Wilson and Roach [21] presented the difficulties in obtaining an optimal stowage plan using a single-phase approach, since it is demonstrated as an NP-hard problem. Thus, Ambrosino et al. [22,23] designed a multi-step heuristic integrated with mathematical programming and another classic algorithm, and Cruz-Reyes et al. [24] proposed a constructive heuristic. Some scholars designed heuristics to generate the entire stowage plan. Dubrovsky et al. [25], Azevedo et al. [26] and Lee and Low [27] added a stability constraint in the objective function as a penalty, and then improved a genetic algorithm using the form of a compact solutions encoding; Azevedo et al. [26] and Araújo et al. [28] both established a bi-objective model to balance the rehandles and stability, which was solved by a variety of heuristics. Monaco et al. [29] considered the quay crane operation, horizontal transport time and rehandling operation in the yard as the terminal operation objectives. These authors’ studies were based on a pre-stowage plan, connecting the ship with the yard, but not considering the interactive influence between the containership stowage plan and the yard operation plan.

In recent years, some scholars have begun to pay attention to the relationship between the containership stowage plan and the yard operation plan. Focusing on the loading operation process of a single ship bay, Zhu et al. [30] defined the block relocation and loading problem (BRLP) in detail and established an integer linear programming model. Considering that the BRLP is a special case of the BRP, a rule-based heuristic algorithm that can be commonly used to solve the BRLP and BRP is designed. Jovanovic et al. [31] further proposed a two-stage greedy algorithm, first selecting the loading container and then proposing the correction process to optimize the rehandling position of the blocked container. Tanaka and Voss [32] defined the unrestricted and restricted block relocation problem based on the ship stowage plan and constructed a general iterative deepening branch and bound algorithm to solve this problem. Existing studies are based on the premise that the ship’s stowage plan is known. Some scholars focus on the same problem as discussed in this study, but the containership has a rectangular format, and all containers have the same weight in Junqueira et al. [33]. In this study, the containership has the practical configuration with the irregular format and the model considers the weight of each individual container and the stability constraints by limiting the range of related indicators, thus having the broader scalability in the real world compared with the existing literature. Actually, stowage planning is a common problem faced by two interacting decision makers, the ship side and the terminal side. The ship side makes decisions at the planning level and gives pre-loading instructions to allocate different types of containers based on different destinations, types, sizes and weights, among other attributes; the terminal side makes more detailed decisions at the operational level and determines the specific storing positions of containers to be loaded on the ship according to the pre-loading instructions received from the ship side. Making the operation plan or storage plan of the yard is a staged decision-making process, and the interaction between the two is not considered. The current stowage system can only achieve the objective of one of decision maker.

To the best of our knowledge, the operation efficiency of the container terminal is largely affected by the yard handling operations and the containers’ distribution on the containership. Our main contribution is to propose a proper methodology to integrate the containership stowage plan and the terminal yard operation plan from the perspective of the terminal system.

This paper firstly details the problem in Section 2, then formulates the integer programming in Section 3. To obtain the optimal solutions, a two-stage optimization algorithm is proposed in Section 4. Section 5 demonstrates the performance of stowage plans where the two-stage algorithm is compared with the results of the mathematical programming for the same problem. Section 6 presents the conclusions.

2. Integrated Containership Stowage Problem

The containership stowage problem (CSP) aims to organize the containers; distribution in the containership and make stowage plans, ensuring ship stability in each port of call. The operational efficiency of container terminals is affected by yard configuration and the container moving plan. The stowage plan can be described as the containers’ distribution in the containership, which should be performed simultaneously by the ship side and the terminal side. The stowage plan must be determined within a limited time period; thus, the configurations of various yard bays cannot be considered by the ship side. The largest containerships would require thousands of container movements at each port of call, causing low operational efficiency in the terminal yard. In conclusion, most stowage plans are currently made to meet the unloading efficiency in the subsequent ports and the force constraints in the containerships’ navigation, but do not care about the operational efficiency of the yard.

The yard’s operational efficiency is measured by the number of movements, including removal loading and removal relocation. A removal loading movement would be generated for each container to be loaded to the ship. Thus, the number of removal loadings cannot be avoided. To improve the terminal yard’s operational efficiency, attention should be paid to the minimization of unproductive container relocations in the storage area. The containers, stacked in a yard bay, are transported quayside and loaded to the ship. The yard bay in this study is comprised of multiple horizontal stacks and vertical tiers, corresponding to a two-dimensional stock space. Containers can be accessed only from the top tier to the bottom tier of the stack. The container to be loaded, available to the containership, is defined as the target container. Any container stacked on the target container must be relocated to other stacks, producing unproductive movements. Industry practitioners focus on the minimization of such unproductive movements, referred to as relocations. The relocation would be caused by shifted location, which has already been established as the container/block relocation problem (CRP/BRP). Moreover, the number of relocations would also be affected by the loading sequence, which was studied in [31], defined as the block relocation and loading problem (BRLP).

This study combines the CSP and BRLP together, that is, the integrated containership stowage problem (ICSP). It may be defined as follows: given the containers’ distribution in the yard layout, the ICSP aims to minimize the relocations in the terminal yard by removing all containers from the storage area and loading them to the ship slots, considering the appropriate stability strength and minimum number of shifts generated in the unloading process in the subsequent ports of the shipping route.

A container terminal layout is illustrated in Figure 1, generally including a quayside area for containerships’ berthing and a terminal yard for the temporary storage of containers. The movements of containers between the containerships and the terminal yard constitute the loading and unloading operations of the containerships. This study focuses on the loading operations, making a plan for containership stowage and terminal yard operation, not considering the scheduling of cranes in the container terminal to simplify the combinatorial optimization problem.

The loading operation is performed by removing a certain container on the highest tier of a yard stack, making sure the corresponding ship stack is filled from the available bottom tier. The input parameters are the number of bays, stacks and tiers of the containership, together with the attributes of the containers to be loaded, which include the size, weight, destination and the storage position in the yard space. The standard size of all containers considered in this paper is 20 TEU, and each yard stack stores containers with the same destination.

Figure 2a illustrates a feasible ship stowage plan considering the destination port and gross weight of each container to be loaded and the physical constraints of weight distribution in the containership. Figure 2b shows the containers’ storage locations in the yard space. The containers marked by the same background pattern have the same affiliations, such as destination port and gross weight. Thus, based on the desired ship stowage plan, containers with the same affiliations can be exchanged in the ship slots, still maintaining the stability of the containership. To maximize the operation efficiency in the yard, and also meet the requirements of containership navigation, containers 1 and 6, or containers 2 and 5, are changed, thus generating the optimal ship stowage plan by reducing the number of relocations to 0 in Figure 2c,d. Thus, the optimal stowage plan designed by considering the containers’ stacking locations in the yard space improves the terminal’s operational efficiency and achieves double wins for the terminal and ship.

The coordination of the ship stowage plan and yard space configuration takes one more step forward compared to the general containership stowage problem. Obviously, the study of the coordinated stowage plan optimization protects the interests of both sides and it can simplify the stowage process, where the repeated confirmations between two sides can be omitted.

Currently, the capacity of the largest containership in the world is up to 20,000 TEUs, including several dozen 20 TEU ship bays. It is more complicated to make stowage plans for a number of ports on a shipping route. Even in a port of call, it is also a large-scale problem, requiring a state-of-the-art computerized method and technology.

3. Mathematical Model

An integer linear program (ILP) is established to model the ICSP, denoted by ICSP-ILP. The model in this study achieves the integration of the CSP and BRLP models, which is not just a combination of both. The integer program is presented in

Table 1. Indices involved in parameters and decision variables of the ICSP-ILP.

Indices	Meaning	Range
c	Serial number of containers	c∊{1,2,…,C}, integer
s	Serial number of yard stacks	s∊{1,2,…,S}, integer
n	Serial number of operations	n∊{1,2,…,N}, integer
g	Serial number of ship stacks	g∊{1,2,…,G}, integer
t	Serial number of ship tiers	t∊{1,2,…,H}, integer
i	Serial number of ship bays	i∊{1,2,…,I}, integer

,

Table 2. Parameters involved in the ICSP-ILP.

Parameters	Meaning
C	Number of containers to be loaded into the ship
S	Number of yard stacks involving containers to be loaded into the ship
N	Number of operations by finishing the loading process
G	Number of ship stacks with empty slots
H	Number of ship tiers with empty slots
I	Number of ship bays with empty slots
mxHeight	Maximum allowed stacking height for any yard stack which is measured by number of containers
mxNum	The large number
initialYardcs	=1 if container c is initially located in yard stack s before the loading process
initialBuryc	Number of containers stacking over container c (including container c itself) in the yard stack before the loading process
Egt	=1 if it is empty of ship slot (g, t)
dc	The destination port of container c
wc	The weight of container c
VDt	The vertical distance between the center of the tth ship tier and hull bottom
HDi	The horizontal distance between the center of the ith ship bay and the hull medium
Kgi	=1 if ship stack g belongs to the ith ship bay
LBg	The minimum ship tier number of the gth ship stack (referring to the ship tier number of the highest slot)
UBg	The maximum ship tier number of the gth ship stack (referring to the ship tier number of the lowest slot)
GM0	The given lower bound of the allowed metacentric height
GM1	The given upper bound of the allowed metacentric height
KM	The vertical distance between the transverse metacentric and the base line
T0	The given lower bound of the allowed trim
T1	The given upper bound of the allowed trim
MTC	Centimeter trim moment
Wj0	The weight of the containership’s constant Pj, including the crew and supplies, spare parts, oil and water
hj	The horizontal distance between the center of gravity of Pj and the midship
vj	The vertical distance between the center of gravity of Pj and the base line
CCGj	The horizontal distance between the center of gravity of Pj and the midship
coe	The coefficient of the hull buoyancy lever (ship manufacturing parameters)
Mom	The total moment of the weight in the first half hull and that in the second half hull (ship manufacturing parameters)
Dis	The displacement of the unloaded containership (ship manufacturing parameters)
L	The length of the containership (ship manufacturing parameters)

,

Table 3. Decision variables in the ICSP-ILP.

Decision Variables	Meaning
Xcsn	=1 if container c has been located in yard stack s before the nth operation
Bcn	Number of containers stacking over container c (including container c itself) in its yard- stack before the nth operation. This is assigned to 0 if container c is removed from the yard at any operation before the nth operation
Ccn	=1 if container c moves (one step) closer to the highest located container of its yard stack during the nth operation
Fcn	=1 if container c moves (one step) farther from the highest located container of its yard stack during the nth operation
Rcsn	=1 if container c is removed from yard stack s during the nth operation
rsn	=1 if a container is removed from yard stack s during the nth operation
Pcsn	=1 if container c is relocated to yard stack s during the nth operation
psn	=1 if a container is relocated to yard stack s during the nth operation
Pgt0	=1 if the container in ship slot (g, t) would be shifted within the unloading process
Acgtn	=1 if container c has been loaded to ship slot (g, t) before the nth operation
agtn	=1 if a container has been loaded to ship slot (g, t) before the nth operation
Ogtn	=1 if ship slot (g, t) has been occupied by a container before the nth operation

and

Table 4. Mathematical formulation ICSP-ILP.

Objective: $\min \mathsf{\alpha }·{\displaystyle {\displaystyle {\displaystyle \sum }_{c=1}^C}}{\displaystyle {\displaystyle \sum }_{s=1}^S}{\displaystyle {\displaystyle \sum }_{n=1}^N}{P}_{csn}+\mathsf{\beta }·{\displaystyle {\displaystyle \sum }_{g=1}^G}{\displaystyle {\displaystyle \sum }_{t=1}^H}{P}_{gt}^0$ or	(1)
$\min \mathsf{\alpha }·{\displaystyle {\displaystyle \sum }_{s=1}^S}{\displaystyle {\displaystyle \sum }_{n=1}^N}{p}_{sn}+\mathsf{\beta }·{\displaystyle {\displaystyle \sum }_{g=1}^G}{\displaystyle {\displaystyle \sum }_{t=1}^H}{P}_{gt}^0$
Subject to:
${\displaystyle {\displaystyle \sum }_{s=1}^S}{r}_{sn}\le 1,\forall n$	(2a)
${\displaystyle {\displaystyle \sum }_{s=1}^S}{r}_{sn}=1,\forall n from 1 to C$	(2b)
${\displaystyle {\displaystyle \sum }_{c=1}^C}{R}_{csn}=r_{sn},\forall s,n$	(2c)
${\displaystyle {\displaystyle \sum }_{c=1}^C}{P}_{csn}=p_{sn},\forall s,n$	(2d)
$P_{csn}+P_{cs\left(n+1\right)}\le 1,\forall c,s,n from 1 to N-1$	(2e)
$p_{sn}+r_{sn}\le 1,\forall s,n$	(2f)
${\displaystyle {\displaystyle \sum }_{g=1}^G}{\displaystyle {\displaystyle \sum }_{t=1}^H}{\displaystyle {\displaystyle \sum }_{n=1}^N}{A}_{cgtn}=1,\forall c$	(3a)
${\displaystyle {\displaystyle \sum }_{c=1}^C}{\displaystyle {\displaystyle \sum }_{n=1}^N}{A}_{cgtn}\le E_{gt},\forall g,t$	(3b)
${\displaystyle {\displaystyle \sum }_{c=1}^C}{A}_{cgtn}=a_{gtn},\forall g,t,n$	(3c)
${\displaystyle {\displaystyle \sum }_{c=1}^C}{\displaystyle {\displaystyle \sum }_{t=1}^H}{A}_{cgtn}={\displaystyle {\displaystyle \sum }_{t=1}^H}{a}_{gtn},\forall g,n$	(3d)
${\displaystyle {\displaystyle \sum }_{g=1}^G}{\displaystyle {\displaystyle \sum }_{t=1}^H}{A}_{cgtn}+{\displaystyle {\displaystyle \sum }_{s=1}^S}{P}_{csn}={\displaystyle {\displaystyle \sum }_{s=1}^S}{R}_{csn},\forall c,n$	(4a)
${\displaystyle {\displaystyle \sum }_{g=1}^G}{\displaystyle {\displaystyle \sum }_{t=1}^H}{a}_{gtn}+{\displaystyle {\displaystyle \sum }_{s=1}^S}{p}_{sn}={\displaystyle {\displaystyle \sum }_{s=1}^S}{r}_{sn},\forall n$	(4b)
$X_{cs1}=initialYar{d}_{cs},\forall c,s$	(5a)
${\displaystyle {\displaystyle \sum }_{s=1}^S}{X}_{csn}\le 1,\forall c,n from 1 to N+1$	(5b)
${\displaystyle {\displaystyle \sum }_{c=1}^C}{X}_{csn}\le mxHeight,\forall s,n from 1 to N+1$	(5c)
$X_{cs\left(n+1\right)}=X_{csn}+P_{csn}-R_{csn},\forall c,s,n$	(5d)
$B_{c1}=initialBur{y}_c,\forall c$	(6a)
$0\le B_{cn}\le mxHeight,\forall c,n from 1 to N+1$	(6b)
$B_{c\left(n+1\right)}=B_{cn}+F_{cn}-C_{cn},\forall c,n$	(6c)
$R_{csn}\le X_{csn},\forall c,s,n$	(7)
$\left(1-{\displaystyle {\displaystyle \sum }_{s=1}^S}{R}_{csn}\right)·\left(mxHeight-1\right)\ge B_{cn}-1,\forall c,n$	(8)
${\displaystyle {\displaystyle \sum }_{c=1}^C}{C}_{cn}\le mxHeight,\forall n$	(9a)
${\displaystyle {\displaystyle \sum }_{c=1}^C}{F}_{cn}\le mxHeight,\forall n$	(9b)
${\displaystyle {\displaystyle \sum }_{n=1}^N}(C_{cn}-F_{cn})=initialBur{y}_c,\forall c$	(9c)
$X_{csn}\ge C_{cn}+r_{sn}-1,\forall c,s,n$	(10a)
$r_{sn}\ge X_{csn}+C_{cn}-1,\forall c,s,n$	(10b)
$C_{cn}\ge r_{sn}+X_{csn}-1,\forall c,s,n$	(10c)
$X_{cs\left(n+1\right)}\ge F_{cn}+p_{sn}-1,\forall c,s,n$	(11a)
$p_{sn}\ge X_{cs\left(n+1\right)}+F_{cn}-1,\forall c,s,n$	(11b)
$F_{cn}\ge p_{sn}+X_{cs\left(n+1\right)}-1,\forall c,s,n$	(11c)
$O_{gt1}=0,\forall g,t$	(12a)
$O_{gt\left(n+1\right)}=O_{gtn}+a_{gtn},\forall g,t,n$	(12b)
${\displaystyle {\displaystyle \sum }_{n=1}^N}n·a_{gtn}+\left(1-O_{gt\left(N+1\right)}\right)·N\ge {\displaystyle {\displaystyle \sum }_{n=1}^N}n·a_{g\left(t+1\right)n},\forall g|U{B}_g-L{B}_g>0,\forall t from L{B}_g to U{B}_g-1$	(12c)
${\displaystyle {\displaystyle \sum }_{c=1}^C}{\displaystyle {\displaystyle \sum }_{n=1}^N}{d}_c·A_{cg{t}_{1}n}-{\displaystyle {\displaystyle \sum }_{c=1}^C}{\displaystyle {\displaystyle \sum }_{n=1}^N}{d}_c·A_{cg{t}_{2}n}-M·\left(1-{\displaystyle {\displaystyle \sum }_{c=1}^C}{\displaystyle {\displaystyle \sum }_{n=1}^N}{A}_{cg{t}_{1}n}\right)\le M·P_{g{t}_1}^0,\forall g|U{B}_g-L{B}_g>0,\forall t_1 from L{B}_g to U{B}_g-1,\forall t_2 from t_1+1 to U{B}_g$	(12d)
$G{m}_0\le KM-\frac{{{\displaystyle \sum }}_{c=1}^C{{\displaystyle \sum }}_{t=1}^H{{\displaystyle \sum }}_{g=1}^G{{\displaystyle \sum }}_{n=1}^N{w}_c·A_{cgtn}·V{D}_t+{{\displaystyle \sum }}_{j=1}^J{w}_j^0·h_j}{{{\displaystyle \sum }}_{c=1}^C{w}_c+{{\displaystyle \sum }}_{j=1}^J{w}_j^0}\le G{m}_1$	(12e)
$T_0\le \frac{{{\displaystyle \sum }}_{c=1}^C{{\displaystyle \sum }}_{i=1}^I{{\displaystyle \sum }}_{t=1}^H{{\displaystyle \sum }}_{g=1}^G{{\displaystyle \sum }}_{n=1}^N{w}_c·A_{cgtn}·H{D}_i·K_{gi}+{{\displaystyle \sum }}_{j=1}^J{w}_j^0·v_j-\left({{\displaystyle \sum }}_{c=1}^C{w}_c+{{\displaystyle \sum }}_{j=1}^J{w}_j^0\right)·LCB}{MTC}\le T_1$	(12f)
$S_0\le 0.5·\left[mom·L·dis+{\displaystyle {\displaystyle \sum }_{c=1}^C}{\displaystyle {\displaystyle \sum }_{i=1}^I}{\displaystyle {\displaystyle \sum }_{t=1}^H}{\displaystyle {\displaystyle \sum }_{g=1}^G}{\displaystyle {\displaystyle \sum }_{n=1}^N}{w}_c·A_{cgtn}·K_{gi}·CC{G}_i+{\displaystyle {\displaystyle \sum }_{j=1}^J}{w}_j^0·CC{G}_j-coe·L·\left({\displaystyle {\displaystyle \sum }_{c=1}^C}{w}_c+{\displaystyle {\displaystyle \sum }_{j=1}^J}{w}_j^0\right)\right]\le S_1$	(12g)

: Table 1, Table 2 and Table 3 list the indices, input parameters and decision variables involved in the mathematical formulation; and finally, the ICSP-ILP is presented in Table 4.

To simplify the ICSP, there are no 40 ft containers and refrigerated or hazardous containers to be loaded into the containership. The containers to be loaded into the containership are all 20 ft general containers.

The ICSP is illustrated by removing some containers from their located yard stacks, and loading into designated ship stacks, involving the containership side and yard side. The containership and yard are both three-dimensional space structures, which are comprised of bays, stacks and tiers. To simplify the ICSP-ILP, this study has to reduce the dimensions of the mathematical model. For the containership, stack and tier are reserved as the two dimensions, where the ship tier dimension is reserved to measure the stability. The ship bay dimension is removed and distinguished by the ship stacks numbered by incrementing integers. Moreover, a specific domain of the ship stacks was defined for each ship bay. A_cgtn and O_gtn are designed to track the initial and dynamically changing status of ship tier storage in the loading process. For the yard, only the stack is reserved. The yard tier dimension is removed from the 3-dimensional structure. However, X_csn and B_cn are designed to track the initial and dynamically changing status of yard tier storage for each yard stack in the operation process. The yard bay dimension is also removed from the formulation, but it can be identified by the serial number of yard stacks numbered by incrementing integers. Thus, a specific domain of the yard stacks is defined for each yard bay. The setting of the decision variables has some implicit meanings. For example, R_csn and r_sn imply that the removed container must be the highest located container in yard stack s before the nth operation; moreover, P_csn and p_sn imply that the relocated container would be the highest located container of yard stack s after the nth operation.

For the container terminal’s sustainable development, the mathematical model was established to reduce the relocations in the yard and the shifts in the containership, simultaneously achieving an optimal loading sequence from the yard to the containership and generating an optimal stowage plan to meet the seaworthiness requirements. The objective function is numbered by (1), in which $\mathsf{\alpha }$ and $\mathsf{\beta }$ are the weight of the relocations in the yard and the shifts in the containership, respectively. For the convenience of the introduction of the ICSP-ILP, this study divides the constraints of the ICSP-ILP into three groups.

The first group from constraints (2) to (8) describes the relationship of the decision variables. Constraint (2a) limits that no more than one container is removed from a yard stack for each operation. Constraint (2b) ensures that there is one and only one container to be removed from a yard stack for the first C operations. Constraint (2c) defines the proper relationship among R_csn and r_sn. Constraint (2d) defines the proper relationship among P_csn and p_sn. Constraint (2e) defines that a container cannot be relocated in consecutive operations. Constraint (2f) ensures a relocated container is relocated to a different yard stack where it is initially located for any operation. Constraint (3a) states that all containers would be loaded to the ship after the loading operation. Constraint (3b) shows that the containers can only be loaded to the existing ship slots. Constraints (3c) and (3d) connect the relationship between A_cgtn and a_gtn. Constraints (4a) to (4b) define the proper relationship between R_csn, r_sn, P_csn, p_sn, A_cgtn and a_gtn. Constraint (5a) initializes X_cs1 according to the initial configuration of containers in the yard space. Constraint (5b) states that there are only two places to go for each container before each operation, which are the yard slot or the ship slot. Constraint (5c) limits the maximal stacking height of containers in a yard stack. Constraint (5d) updates X_csn for each operation if there is any relocation or removal in the yard stack at that operation. Constraint (6a) initializes B_c1 with proper values. Constraint (6b) limits the value ranges of B_cn. Constraint (6c) updates B_cn for each operation if there are containers getting further away or closer to the container in the highest tier of the corresponding yard stacks at the operation. Constraint (7) limits that any container can only be removed from the yard stack where the container is located currently. Only containers in the highest tier of yard stacks can be removed, ensuring each container is stacked on another container, except the container in the lowest tier of a yard stack. Constraint (8) restricts that the removal operations take place on containers which are located in the highest tier of yard stacks, and it succeeds in avoiding the suspending in the yard stack.

The second group from Constraints (9) to (11) lists the implicit constraints about C_cn and F_cn. Constraint (9a) or (9b) allows the maximal number of container movements (closer to/father from) the containers in the highest tier of the corresponding yard stacks at each operation. Constraint (9c) limits the exact number of movements closer to the container in the highest tier of a yard stack for each container in the loading process. Constraints (10a) to (10c) link C_cn to X_csn and r_sn, which state that if any two of X_csn, C_cn and r_sn are equal to 1, then the third is also equal to 1. Constraints (11a) to (11c) link F_cn to X_cs(n+1) and p_sn, which state that if any two of X_csn, F_cn and p_sn are equal to 1, then the third is also equal to 1.

The third group ensures the ship meets the seaworthiness requirements. Constraint (12a) initializes O_gtn, since the slot in ship stack g and ship tier t is empty initially. Constraint (12b) updates O_gtn for each operation if there is a container to be loaded to the yard stack at that operation. The constraint makes sure the container in the lower ship tier should be loaded to the ship stack earlier than that in the higher ship tier. Actually, containers should be loaded to each ship stack from the lower ship slot to the higher ship slot, ensuring each container is stacked on another container, except the container at the bottom tier of a ship stack, and avoiding suspension in the ship stack, which is ensured by Constraint (12c). In the real containership, there are several kinds of space structures with different ship stacks for each ship bay and different ship tiers for each ship stack, which increases the difficulty in the model building. Due to the irregularity of the ship bays, parameters UB_g, LB_g and K_gi are added into the model to make the Constraint (12c,d) expressed easily. Constraint (12d) is used to justify whether a container should be shifted in the containership within the unloading process. Constraints (12e–g) ensure the containership’s reasonable stability, trim and strength, respectively, after all containers are loaded to the containership.

4. Solutions

The difficulties in optimizing the large-scale instances in the real world by directly solving the mathematical program encourage us develop heuristic algorithms to arrive at a solution for the ICSP, merging the problem feature and human experience into the computer language. A rule-based heuristic is proposed, in which Stage 1 produces an assignment plan, illustrating the loading sequence and the corresponding relationship between the container groups and ship blocks. After that, the assignment plan is improved by genetic evolution to meet the seaworthiness constraints. In Stage 2, the detailed stowage plan is generated to cut down the relocations in the terminal yard, including the stowage document, the container loading sequence and the yard rehandling operation.

4.1. Generate the Initial Stowage Plan (S₁)

4.1.1. Container Group Allocation Scheme

A heuristic algorithm HAF is proposed in Stage 1 to make a feasible stowage plan to minimize the shifts on the ship when discharging containers, ignoring the stability–strength considerations. A ship bay is divided into two blocks, which is helpful to find the feasible solutions. Containers, grouped by the destination from the farthest to the nearest, are loaded to the ship blocks randomly if no shifting occurs on the ship when unloading containers. In this stage, an assignment plan is generated to connect the container groups and ship blocks.

The pseudocode is used to describe the HAF, and the expressions in the pseudocode are defined in

Table 5. Expressions newly appearing in the pseudocode of the HAF.

Expressions	Definition
b	Index of the blocks, partitioned by the ship bay configuration
d	Index of the ports by consecutive numbers based on a ship’s visiting order on a given route
dmax	The farthest port number
dmin	The nearest port number
Numd	The number of containers with destination d should be loaded to the ship
Block_E	The set of the empty block
Block_N	The set of the block with some containers but not full
DPb	Serial number of the nearest port of containers in block b
Spaceb	Quantities of empty slots in block b
weightgt	The weight of container in ship slot (g, t)
origingt	Serial number of starting port of the container in ship slot (g, t)
destinationgt	Serial number of destination port of the container in ship slot (g, t)
Weight	The stowage plan for the weight distribution (the matrix for weightgt)
Origin	The stowage plan for the origin port distribution (the matrix for origingt)
Destination	The stowage plan for the destination port distribution, which is a matrix consisting of destinationgt for any ship slot (g, t)
AP	The assignment plan

.

Here lists the pseudocode of the HAF (Algorithm 1):

Algorithm 1. HAF

Input dmax, dmin, Numd, Block_E, Block_N, DPb, Spaceb, weightgt, origingt, destinationgt, Weight, Origin, Destination, AP For d = dmax:−1: dmin      While Numd > 0           Choose a block b from the Block_E∪Block_N randomly           If b∈Block_E                If Numd ≥ Spaceb                    Numd ←Numd−Spaceb                    Spaceb ← 0                Else                    Spaceb ←Spaceb−Numd                    Numd ← 0                End If                Block_E ← Block_E−b                DPb ← d           Else                If DPb ≤ d                     If Numd ≥ Spaceb                         Numd ←Numd−Spaceb                         Spaceb ← 0                     Else                         Spaceb ←Spaceb−Numd                         Numd ← 0                         Block_N ← Block_N−b                     End If                     DPb ← d                End If                AP ← [AP;b, d, Spaceb]                Update weightgt, origingt, destinationgt, Weight, Origin, Destination           End While      End For Output Numd, Block_E, Block_N, DPb, Spaceb, weightgt, origingt, destinationgt, Weight, Origin, Destination, AP

Finally, an assignment plan is illustrated by a three-row matrix, as illustrated in Figure 3, in which the first row refers to the block number, the second row refers to quantities of containers assigned to the block and the third row refers to the destination port of the assigned containers. Moreover, the column number of the matrix represents the loading order of blocks.

4.1.2. Container Location Adjustment

Considering the large scale of the ICSP and the scalability of the genetic algorithm, a genetic algorithm GA is developed to optimize the initial feasible assignment plan meeting the seaworthiness requirements of stability, strength and trim. It performs outstandingly using group search features to effectively avoid searching for points that do not need to be searched to speed up the solution. Moreover, the algorithm can be easily revised to apply to other settings.

The GA is developed to optimize the initial stowage document generated from the first stage. The stowage document consists of three real matrices illustrating the ship slots’ configuration. The three real matrices refer to the weight, origin port and destination port of the container in the corresponding ship slots, and they are represented by weight, origin and destination, respectively.

The key process of the GA includes selection, crossover and mutation. Here lists the main design method for the key processes of the GA:

• Selection and crossover

  Step 1

  Choose randomly a destination port from the third row of the matrix representing the assignment plan and find the corresponding blocks storing the containers with the destination port.

  Step 2

  If there are at least two blocks, then choose b₁ and b₂ from them randomly; otherwise, go back to Step 1.

  Step 3

  Choose part of the containers in a block b₁ and the same number of containers in another block b₂. These containers are all loaded to the ship from the current port, and they will go to the same destination port.

  Step 4

  Exchange these containers from two blocks b₁, b₂, and make the recombination of the new assigned container in the two blocks.

  Step 5

  Generate the new stowage plan.

Figure 4 gives an example for the selection and crossover step, in which the assignment plan stays the same.

• Selection and mutation

  Step 1

  Choose some containers from a block b₁, and these containers are loaded to the ship from the current port, and they go to the same destination port d; and choose a non-full block b₂, but the nearest destination port of the loaded containers in the block must be further than d.

  Step 2

  Remove these containers from the block b₁ and load to the non-full block b₂, and then make the recombination of block b₁ and b₂.

  Step 3

  Generate the new stowage plan.

Figure 5 gives an example of the selection and mutation step, in which a task of the assignment plan must be revised, and another new task may be added to the assignment plan.

Accepting the new stowage plan or not depends on whether the new stowage plan achieves more reasonable stability, strength and trim in meeting the seaworthiness requirements.

4.2. Formulate the Optimal Stowage Plan and Loading Plan (S₂)

Finally, a heuristic algorithm (HAD) is designed to make a detailed stowage plan including the stowage document (weight, container, origin and destination), loading sequence and rehandling operation.

Some expressions listed in

Table 6. Expressions newly appearing in the HAD flow chart.

Expressions	Definition
SW	The assigned weight set of each lowest ship stack slot
MS	The yard stack set with different weight level
MW	The weight set of each highest yard stack slot in MS
PS	The yard stack set with completely same weight level
PW	The weight set of each yard stack in PS
Containergt	The container number in the tth ship tier of the gth ship stack
Container	The stowage plan of the container number (the matrix for Containergt)
LS	Set of lowest containers in each ship stack
TY	Set of tier’s serial number for LS in the corresponding yard stacks
EY	Set of the current stacking height of corresponding yard stacks
RS	Set of feasible yard- stacks for relocation

are defined before introducing the HAD.

The basic idea of the HAD is to firstly load containers with the corresponding weight and destination port to the ship slots from the yard stack’s top tier. If there is no container with the corresponding weight and destination port in the yard stack’s top tier, an adjustment is made to the original stowage plan for the weight distribution, to accomplish the loading operation without relocations in the yard. If the seaworthy requirements of the adjusted stowage plan conflict with the design value, the original stowage plan remains and rehandling operations are generated in the yard. Figure 6 describes the flow chart of the HAD.

In rehandling operations, the container buried by the minimal number of containers is selected as the target container, which is regarded as the container to be loaded to the ship. Before the retrieval of the target container, the containers stacking over the target container must be removed from the target container and relocated to other yard stacks for temporary storage. The empty yard stack is selected for relocation in the first priority. If there is no empty yard stack currently, thelowest yard stack is chosen for relocation.

The rehandling operation is a little complicated, so it is differentiated from the flow chart of the HAD, and a detailed description is made of the rehandling decision process, illustrated in Figure 7.

In summary, the HAD is the framework of the second stage, mainly focusing on the stowage plan, while the rehandling operation is part of the HAD, deciding on the loading sequence from the yard to the ship based on the stowage plan. The interaction of the stowage plan and the loading sequence is controlled by the proposed rules in the HAD and measured by the relocations generated in the ship by Figure 6 and the rehandles in the yard by Figure 7.

5. Numerical Experiments

Table 7. Comparisons of the results by CPLEX, GUROBI and MATLAB.

No.#: P × C	CPLEX		GUROBI		MATLAB
	Objective	Runtime (s)	Objective	Runtime (s)	Objective	Runtime (s)
1: 2 × 6	0	0.1	0	0.2	0	<0.1
2: 2 × 6	0	0.1	0	0.2	0	<0.1
3: 2 × 6	1	0.5	1	0.3	1	<0.1
4: 4 × 12	0	0.2	0	8.4	0	<0.1
5: 4 × 12	0	0.2	0	8.1	0	<0.1
6: 4 × 12	1	514.2	1	569.0	1	<0.1
7: 4 × 18	0	195.0	0	17.6	0	<0.1
8: 4 × 18	0	232.0	0	16.8	0	<0.1
9: 4 × 24	0	14,470.6	0	2293.4	0	<0.1
10: 4 × 24	0	5836.5	0	4298.0	0	<0.1

lists the comparisons of the results by CPLEX, GUROBI and MATLAB for 10 small-scale instances. In the No.#: P × C column, P represents the quantity of ports of call in the ship voyage; C represents the quantity of containers to be loaded to the ship in the current port; No.# indicates the instance number. The objective values achieved by CPLEX and GUROBI must be the optimal solution. It is easily seen that the three-stage method coded in MATLAB can also reach the lower bound in all listed instances. However, only small-scale instances can be solved by CPLEX and GUROBI within the accepted time because of the largest number of variables and constraints in the mathematical programming model. The two-stage algorithm is more effective in solving the ICSP. All instances are available in the repository: https://github.com/Ivygood/instances (accessed on 18 March 2022).

To investigate the performance of the continuous procedure in the two-stage method, computational experiments are made on 10 kinds of instances, which are the case combinations of P and C. Assuming that there are 700, 800, 900, 1000 or 1100 TEU containers to be loaded to the ship, which would call at four or five ports, these parameter values are selected for instance generation, reflecting the real stowage planning when this kind of containership is put into operation under different shipping lines. A total of 10 instances are generated for each case with different weight combinations.

Table 8. The results by the rule-based algorithm.

No.#: P × C	S1					S2				Time (s)
	Tr.	Sta.	Str.	Obj.	Rate	Tr.	Sta.	Str.	Obj.
1: 4 × 700	0	2.31	9.85	83	11.86%	−0.15	2.25	9.53	0	26.86
2: 4 × 800	0	2.04	9.69	85	10.63%	−0.12	1.97	9.48	0	35.50
3: 4 × 900	0	1.94	6.32	58	6.44%	0.06	1.92	6.33	0	34.05
4: 4 × 1000	0	2.12	7.69	32	3.20%	−0.05	2.06	7.71	0	35.24
5: 4 × 1100	0	2.01	6.26	62	5.64%	−0.11	1.81	6.40	0	52.33
6: 5 × 700	0	1.72	8.88	103	14.71%	−0.10	1.62	8.74	0	24.91
7: 5 × 800	0	2.08	7.73	98	12.25%	0.07	2.02	7.77	0	28.24
8: 5 × 900	0	2.10	6.73	126	14.00%	−0.21	2.00	6.56	0	32.60
9: 5 × 1000	0	2.08	8.82	99	9.90%	−0.17	1.97	8.73	0	42.25
10: 5 × 1100	0	1.92	5.74	105	9.55%	−0.19	1.73	5.75	0	48.43

Note: Tr. and Sta. represent the trim and stability of the stowage plan, and the unit is meter (m). Str. represents the strength of the stowage plan, and the unit is 10⁴ ton force each meter (tf/m). Obj. represents the number of rehandles.

lists the average results of 10 instances in the rule-based heuristics, showing the optimization process. For each case, the trim tends to be steady in S₁, while the number of relocations decreases in S₂ with a slight fluctuation of the trim. The optimal solutions are all achieved within 1 min, which is an efficient performance within acceptable limits. There is a large number of iterations in S₁ and S₂, consuming CPU usage.

By investigation from Hutchison Ports YANTIAN, the rehandling rate from the yard side is almost 70% because of the scarce yard space, while the industrial level of rehandling rate from the yard side is almost 15%. Observing the results by the rule-based algorithm, the total rehandling rate from both sides in S₁ is also listed in Table 8, obviously lower than the average level. After the second stage of optimization, the rehandling rate for each case reduces to 0. By comparing with the practical information, the proposed algorithm has a great degree of improvement in the operational efficiency in solving the practical containership stowage problem.

6. Conclusions

This study proposes a methodology for integrating the containership stowage plan and the terminal yard operation plan with the objective of minimizing rehandling operations in the yard. Due to the limitation of computational complexity, the method is described by a two-stage algorithm to complete the entire stowage plan, and after that, to optimize the block stowage plan. The first stage is designed for the fast generation of a feasible solution using a container group allocation scheme to speed up the following optimization; then it develops a genetic algorithm to optimize the initial feasible assignment plan, displaying the representation of the selection, crossover and mutation to make the parent chromosome evolve into the competitive offspring chromosome. The second stage involves the terminal yard in the container stowage problem, making the loading plans corresponding to the containership and container yard. Computational tests show that the proposed algorithm works better than a commercial and established software for solving the integer linear programming model in Section 3. Experimental results show that the proposed algorithm can further help the container terminal maintain a sustainable development process by reducing the relocations in the yard and the shifts in the containership.

This study offers a framework for further assessments of yard handling operations’ impact on the stowage plan. In practice, the yard side and the containership side independently make decisions, ignoring the interactive impact. The integration of both sides in this study makes the decision superior to the practice. Although a lot of practical constraints and specific characteristics of the problem are considered in this study, there are still a lot of practical problems that have been ignored. For example, there are various sizes of containers, but only one standard type of container is considered in this study. If 40 ft and 20 ft containers are to be loaded, 40 ft containers should be placed upon two 20 ft containers, and 20 ft containers cannot be placed upon 40 ft containers because of safety limitations. Moreover, containers are stored in a hold area or on the deck of a containership, separated by the hatch cover. All containers on deck may be unloaded completely from the ship before unloading the containers in the hold, producing a large number of shifts in the ship. This kind of shift caused by hatch covers should be avoided to make an apparent improvement of ship turnaround time, which should also be considered in future extensions of this research.