# Sequential Optimization of a Temporary Storage Location for Cooperative Twin Overhead Shuttles in a Rail-Based Automated Container Terminal

Bonggwon Kang, Bosung Kim, Soondo Hong

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Bonggwon Kang, Bosung Kim, Soondo Hong. Sequential Optimization of a Temporary Storage Location for Cooperative Twin Overhead Shuttles in a Rail-Based Automated Container Terminal. IFIP International Conference on Advances in Production Management Systems (APMS), Sep 2022, Gyeongju, South Korea. pp.285-292, 10.1007/978-3-031-16407-1_34. hal-05386057

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Sequential Optimization of a Temporary Storage
Location for Cooperative Twin Overhead
Shuttles in a Rail-Based Automated Container
Terminal

Bonggwon Kang<fnref n="1" />, Bosung Kim<fnref n="2" />, and Soondo Hong<fnref n="2" />(✉)

**Abstract.** Twin overhead shuttle cranes (OSs) transport containers in a rail-based automated container terminal (RACT). Terminal operators separate a job into a main job and an auxiliary job based on a temporary storage location. Since the temporary storage location determines the frequency of the job separations and the workload of each OS, they use simulation-based decision-making to investigate the impact of the interference between the twin OSs. It is time-intensive to optimize an objective function with manually designed experiments, so this study proposes a sequential optimization approach, Bayesian optimization (BO), to determine the optimal temporary storage location within a limited simulation run. The BO adaptively draws the surrogate model of simulation outcomes and actively suggests the most promising solution comparing the current optimal solution. An experiment demonstrates that the BO predicts the outcomes of a RACT simulation and ensures a near-optimal solution within a limited simulation run.

**Keywords:** Twin overhead shuttles · Bayesian optimization · Rail-based automated container terminal · Simulation

# 1 Introduction

To improve the efficiency and reliability of container handling, Lee, et al. [1] developed a rail-based automated container terminal (RACT). In the RACT, an overhead shuttle (OS) performs a similar function as a yard crane (YC) in a typical container terminal [2]. Flatcars transfer containers between a quay crane and an OS along guided ground rails. The storage yard consists of multiple overhead rails with twin OSs for each rail. Figure 1 illustrates the conceptual layout.

<footnote n="1">¹ Major in Industrial Data Science and Engineering, Department of Industrial Engineering, Pusan National University, Busan 46241, South Korea</footnote>

<footnote n="2">² Department of Industrial Engineering, Pusan National University, Pusan 46241, South Korea
soondo.hong@pusan.ac.kr</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Top view of the twin OS system with container flows in the RACT.</figcaption></figure>

Since the twin OSs in a rail cannot pass each other, terminal operators separate each job into a main job and an auxiliary job based on a temporary storage location. The job separation procedure aims to smooth the synchronization between the twin OSs. The temporary storage location determines the frequency of job separations and the workload of each OS.

The workloads of the OSs are black-box functions due to complex and unknown constraints. Therefore, terminal operators commonly use discrete-event simulations to mimic yard operations, e.g., container handling equipment scheduling, vehicle congestion, interference between yard cranes, and the like, and obtain the optimal decision within a certain planning period [3–6]. However, the expensive objective function to evaluate hinders obtaining an optimal solution within a limited time [7, 8].

In this study, we suggest a sequential optimization approach, Bayesian optimization (BO), to determine the optimal temporary storage location within a limited simulation run. The BO adaptively draws the surrogate model of simulation outcomes, and then actively suggests the most promising optimal solution comparing the current best solution based on an acquisition function controlling the trade-off between exploration and exploitation of searches.

The main contributions of this study are as follows: (1) We investigate the temporary storage location determination problem in a RACT, unlike most studies which mainly treat crane scheduling with a fixed temporary storage location; and (2) We propose a sequential optimization approach, BO, to determine the temporary storage location. A simulation experiment evaluates the proposed model compared to optimal solutions.

This study is structured as follows: Sect. 2 introduces the detailed job separation procedure and the two optimization approaches to minimize the maximum workload of twin OSs in a RACT. Section 3 describes the experiment and the results of the proposed approaches compared to an optimal solution. Section 4 concludes and suggests future research.
# 2 Problem Methodology

## 2.1 Container Flows with a Job Separation Procedure

In the RACT, there are four input/output (I/O) points and three stacking areas (see Fig. 1). A transfer request can be to or from an inbound or outbound container. The inbound (import and unloading) containers move from a vessel to an ET (external truck). The outbound (export and loading) containers move from an ET to a vessel. OSs transport them to and from flatcars at seaside and to and from ETs at landside. Requests for export and unloading containers are storage requests and requests for import and loading are retrieval requests. To complete a transfer request, the OSs repeat four basic operations: empty travel, pick-up, loaded travel, and drop-off.

A job separation procedure is for collaborative operations between the twin OSs whereby one OS transfers a container from its original location to a temporary storage location and the other OS transfers it from the temporary storage location to the container's destination. The job separation procedure balances the workloads of OSs and alleviates interference between twin OSs.

Gharehgozli, et al. [9] showed that using two temporary storages decreases rehandlings, the extra jobs to remove the stacked containers on a target container. We fix the sizes of the temporary storages according to Little's law, $L = \lambda \times W$, where L denotes the estimated size, W denotes the mean dwelling time of stacked containers in the temporary storages, and $\lambda$ denotes the mean arrival rate.

## 2.2 Temporary Storage Location Determination Model

To optimize a temporary storage location, we formulate a temporary storage location determination model (TSLDM) based on MIP. The TSLDM clarifies our optimization problem with linear constraints. We use the following definitions.

### Indices and Sets

$N$ = Set of nodes representing all terminal sections, $N = \{SA_1, SA_2, SA_3, IO_1, IO_2, IO_3, IO_4, B_u, B_l\}$.

$SA$ = Set of nodes representing stacking areas, $SA = \{SA_1, SA_2, SA_3\}$, $SA \subset N$.

$IO$ = Set of nodes representing I/O points, $IO = \{IO_1, IO_2, IO_3, IO_4\}$, $IO \subset N$.

$i, j$ = Index of nodes representing all terminal sections.

$T, t$ = Set of job types and its index, $T = \{U, L, I, O\}$, $t \in T$.

$K, k$ = Set of OSs and its index, $K = \{OS_{sea}, OS_{land}\}$, $k \in K$.

$PU, u$ = Set of unloading storage candidates and its index, $u \in PU$.

$PL, l$ = Set of loading storage candidates and its index, $l \in PL$.

$V, v$ = Set of vessels and its index, $v \in V$.

### Parameters

$F_{vi}^t$ = Number of requests of job type $t$ from node $i$ of vessel $v$.

$UR_v$ = Ratio of unloading operations using a temporary storage of vessel $v$.
$LR_v$ = Ratio of loading operations using a temporary storage of vessel v.

$t_{ijk}$ = Processing time of OS k from node i to j.

$u_{t_{ijk}}$ = Processing time of OS k from node i to j using unloading storage u.

$l_{t_{ijkl}}$ = Processing time of OS k from node i to j using loading storage l.

## Decision Variables

$MF$ = Maximum workload of OSs

$x_u$ = 
$$ \begin{cases} 
1 & \text{If unloading storage u is used,} \\
0 & \text{Otherwise.}
\end{cases} $$

$y_l$ = 
$$ \begin{cases} 
1 & \text{If loading storage l is used,} \\
0 & \text{Otherwise.}
\end{cases} $$

$F_{vijk}$ = Number of container flows of OS k from node i to j except auxiliary flows of vessel v.

$F_{Uvijk}$ = Number of auxiliary flows of OS k from node i to j using unloading storage u of vessel v.

$F_{Lvijkl}$ = Number of auxiliary flows of OS k from node i to j using loading storage l of vessel v.

## Objective function

$$ (\text{TSLDM}) \quad \text{minimize} \quad MF \quad (1) $$

## Subject to

$$ MF \geq \sum_{i \in N} \sum_{j \in N} F_{vijk} \cdot t_{ijk} + \sum_{i \in N} \sum_{j \in N} \sum_{u \in PU} FU_{vijk} \cdot u_{t_{ijk}} \quad \forall k \in K, + \sum_{i \in N} \sum_{j \in N} \sum_{l \in PL} FL_{vijkl} \cdot l_{t_{ijkl}}, \quad (2) $$

$$ F_{vijk} \geq f_{vi}^{U} \cdot (1 - UR_{v}), \quad \forall v \in V, \forall i \in \{IO_{1}\}, \quad \forall j \in \{SA_{1}\}, \forall k \in \{OS_{sea}\}, \quad (3) $$

$$ F_{vijk} \geq f_{vi}^{L} \cdot (1 - LR_{v}), \quad \forall v \in V, \forall i \in \{SA_{1}\}, \quad \forall j \in \{IO_{1}\}, \forall k \in \{OS_{sea}\}, \quad (4) $$

$$ \sum_{j \in SA} \sum_{k \in K} F_{vijk} \geq f_{vi}^{I}, \quad \forall v \in V, \forall i \in IO, \quad (5) $$

$$ \sum_{j \in IO} \sum_{k \in K} F_{vijk} \geq f_{vi}^{O}, \quad \forall v \in V, \forall i \in SA, \quad (6) $$

$$ FU_{vijk} \geq \sum_{w \in N} f_{vw}^{U} \cdot UR_{v} \cdot x_{u}, \quad \forall v \in V, \forall i \in \{IO_{1}\}, \forall j \in \{B_{u}\}, \quad \forall k \in \{OS_{sea}\}, \forall u \in PU, \quad (7) $$

$$ FU_{vijk} \geq \sum_{w \in N} f_{vw}^{U} \cdot UR_{v} \cdot x_{u}, \quad \forall v \in V, \forall i \in \{B_{u}\}, \forall j \in \{SA_{3}\}, \quad \forall k \in \{OS_{land}\}, \forall u \in PU, \quad (8) $$

$$ FL_{vijkl} \geq \sum_{w \in N} f_{vw}^{L} \cdot LR_{v} \cdot y_{l}, \quad \forall v \in V, \forall i \in \{SA_{2}\}, \forall j \in \{B_{l}\}, \quad \forall k \in \{OS_{land}\}, \forall l \in PL, \quad (9) $$

$$ FL_{vijkl} \geq \sum_{w \in N} f_{vw}^{L} \cdot LR_{v} \cdot y_{l}, \quad \forall v \in V, \forall i \in \{B_{l}\}, \forall j \in \{IO_{1}\}, \quad \forall k \in \{OS_{sea}\}, \forall l \in PL, \quad (10) $$

$$ \sum_{u \in PU} x_{u} = 1, \quad (11) $$

$$ \sum_{l \in PL} y_{l} = 1, \quad (12) $$
$$x_u + y_u \le 1, \quad \forall u \in PU. \qquad (13)$$

The TSLDM aims to minimize the maximum workloads of OSs (constraints (1) and (2)). Constraints (3) and (4) calculate the unloading and loading flows without a job separation procedure. Constraints (5) and (6) calculate the import and export flows. Constraints (7), (8), (9), and (10) calculate the unloading and loading flows with job separations considering the temporary storage. Constraints (11) and (12) determine the temporary storage locations. Constraint (13) ensures that the temporary storage locations should not be overlapped.

## 2.3 Sequential Optimization of a Temporary Storage Location

The BO consists of a surrogate function and an acquisition function. We use the Gaussian process (GP) as a surrogate function, which is suitable to take into a noise variance [10]. The GP is identified as a multivariate Gaussian distribution drawn by its kernel function and statistically approximates a posterior distribution of the maximum workloads of the twin OSs according to the decision variables i.e., the locations of temporary storages for loading and unloading requests. Based on the posterior distributions, an acquisition function carefully determines where to sample with a limited number of trials. We use maximum expected improvement (MEI) as an acquisition function to maximize the quality of solutions considering noise measurements. Because the BO treats an objective function with a continuous domain, we apply a naïve rounding approach which rounds the recommended temporary storage locations to the nearest discrete locations at the end [11].

We let $f(x)$ denote an underlying simulation's response over the input variable $x$:

$$f(x) \sim GP(m(x), k(x, x')), \qquad (14)$$

where $f(x)$ is specified by mean function $m(x)$ and covariance function $k(x, x')$. We utilize the most commonly used covariance function, the radial basis function, as kernel function $k$. We derive the posterior distribution for unknown $x_{n+1}$ based on Bayesian statistics [10] as:

$$P(f(x_{n+1}|D_{1:N})) = N(u_n(x_{n+1}), \sigma_n^2(x_{n+1})) \qquad (18)$$

EI (Expected Improvement) provides $x_{n+1}^*$ with the maximum expected improvement by controlling the trade-off between exploration and exploitation of searches according to the posterior distribution:

$$x_{n+1}^* = \operatorname*{argmax}_{x \in \chi} \operatorname{EI}_n(x). \qquad (19)$$

EI quantifies the expected improvement over the input variables toward the optimal decision [12], $\gamma$ denotes the trade-off between exploitation and exploration, $y^+$ denotes the current best observations, $\Phi$ denotes the probability density function, $\Phi$ denotes the cumulative density function, and $Z$ denotes the standard normal distribution. Then,

$$\operatorname{EI}_n(x) = \begin{cases} (u_n(x) - y^+ - \gamma)\Phi(Z) + \sigma_n(x)\phi(Z) & \text{if } \sigma(X) > 0 \\ 0 & \text{if } \sigma(X) = 0' \end{cases} \qquad (20)$$
where

$$Z_n = \begin{cases} \frac{u_n(x)-y^+-y^-}{\sigma_n(x)} & \text{if } \sigma(X) > 0 \\ 0 & \text{if } \sigma(X) = 0 \end{cases}. \quad (21)$$

## 3 Experiment and Results

We build a single-bay simulation model and specify a maritime demand using four factors: the average of loading and unloading containers ($N_{sea}$), the portion of import containers (RI), the portion of unloading containers (RU), and the portion of landside containers (RL). We assume a demand pattern where RI, RU, and RL are 50%, 50%, and 40%, respectively. For example, we can obtain the number of containers from the landside by multiplying RL to $N_{sea}$. $N_{sea}$ are 30, 50, and 70 with different vessel specifications. We randomly generate the different sets of 20 vessels where a vessel requires a uniformly distributed number of transfer requests within the intervals $[0.8 \times N_{sea}, 1.2 \times N_{sea}]$. We observe the cumulative maximum workload of twin OSs over different instances and evaluate the proposed approaches compared with known optimal solutions.

Table 1 summarizes the means and standard deviations of the maximum workloads. The BO with the larger simulation budgets ensured the reduced maximum workload with its small standard deviation. The BO with different simulation budgets ($b = 25, 50, \text{and } 100$) obtained the average optimality gaps as 5.79, 2.99, and 1.03% respectively. The BO with $b = 100$ consistently converged to the near-optimal solution in the range between 0.08 and 1.89%. Notably, the TSLDM obtained the average optimality of 20.79% with a high variation in solution qualities because could not consider the impacts of time-variant interferences among the twin OSs and vehicles.

<table><caption>**Table 1.** Maximum workload and optimality gaps obtained by the TSLDM and the BO.</caption><thead><tr><th rowspan="2">Instance</th><th colspan="5">Maximum workload (hours)</th><th colspan="4">Optimality</th></tr><tr><th>Optimal</th><th>TSLDM</th><th>BO<br>(b = 25)</th><th>BO<br>(b = 50)</th><th>BO<br>(b = 100)</th><th>TSLDM</th><th>BO<br>(b = 25)</th><th>BO<br>(b = 50)</th><th>BO<br>(b = 100)</th></tr></thead><tbody><tr><td>1</td><td>1157.02</td><td>1559.77</td><td>1281.35</td><td>1169.72</td><td>1171.85</td><td>34.81%</td><td>10.75%</td><td>1.10%</td><td>1.28%</td></tr><tr><td>2</td><td>1179.02</td><td>1421.16</td><td>1343.83</td><td>1197.65</td><td>1180.01</td><td>20.54%</td><td>13.98%</td><td>1.58%</td><td>0.08%</td></tr><tr><td>3</td><td>1219.65</td><td>1322.67</td><td>1288.19</td><td>1251.96</td><td>1234.16</td><td>8.45%</td><td>5.62%</td><td>2.65%</td><td>1.19%</td></tr><tr><td>4</td><td>1193.88</td><td>1313.48</td><td>1207.42</td><td>1251.52</td><td>1216.41</td><td>10.02%</td><td>1.13%</td><td>4.83%</td><td>1.89%</td></tr><tr><td>5</td><td>1134.90</td><td>1209.12</td><td>1210.43</td><td>1145.98</td><td>1142.43</td><td>6.54%</td><td>6.66%</td><td>0.98%</td><td>0.66%</td></tr><tr><td>6</td><td>1219.12</td><td>1651.14</td><td>1235.72</td><td>1246.25</td><td>1223.37</td><td>35.44%</td><td>1.36%</td><td>2.23%</td><td>0.35%</td></tr><tr><td>7</td><td>1197.79</td><td>1622.68</td><td>1210.07</td><td>1228.12</td><td>1217.99</td><td>35.47%</td><td>1.03%</td><td>2.53%</td><td>1.69%</td></tr><tr><td>8</td><td>1190.89</td><td>1337.65</td><td>1226.05</td><td>1323.11</td><td>1210.78</td><td>12.32%</td><td>2.95%</td><td>11.10%</td><td>1.67%</td></tr><tr><td>9</td><td>1211.77</td><td>1644.86</td><td>1241.81</td><td>1226.73</td><td>1221.03</td><td>35.74%</td><td>2.48%</td><td>1.23%</td><td>0.76%</td></tr><tr><td>10</td><td>1119.70</td><td>1215.87</td><td>1253.52</td><td>1138.17</td><td>1127.46</td><td>8.59%</td><td>11.95%</td><td>1.65%</td><td>0.69%</td></tr><tr><td>Mean</td><td>1182.37</td><td>1429.84</td><td>1249.84</td><td>1217.92</td><td>1194.55</td><td>20.79%</td><td>5.79%</td><td>2.99%</td><td>1.03%</td></tr><tr><td>Std.v</td><td>32.99</td><td>166.54</td><td>41.32</td><td>53.48</td><td>35.12</td><td>12.42%</td><td>4.62%</td><td>2.91%</td><td>0.58%</td></tr></tbody></table>
# 4 Conclusions

It is time-consuming to run simulations based on manually designed experiments to optimize an underlying objective function. Time is also a factor when significant variations in maritime demands or unexpected breakdowns of equipment occur.

For prompt and effective simulation-based decision-making in yard operations, we propose a sequential optimization model, BO, for determining the location of temporary storages to minimize the maximum workload of twin OSs. The results of our experiment demonstrate that the BO consistently obtains near-optimal solutions, whereas the TSLDM with linear constraints cannot provide the optimal solutions with their high variations.

An integrated analysis of the collaborative operations of twin OSs considering scheduling and routing methods for vehicles merits further research. We suggest that a sophisticated discretization approach could overcome the limitation of the naïve rounding approach used in this study which can cause the BO to repeatedly search previously observed points and become stuck. We also suggest expanding the proposed BO to improve collaborative operations between twin YCs in traditional vertical terminals in the future.

**Acknowledgments.** This work was supported by the 2022 BK21 FOUR Program of Pusan National University and a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. NRF-2020R1A2C2004320).

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