Optimization of Resource Allocation in Automated Container Terminals

Abstract

Automated container terminals have been constructed to reduce emissions and labor cost. Resource allocation problems in automated container terminals have a critical effect on handling efficiency and cost. This paper addresses this problem with quay crane (QC) double cycling in automated container terminals. An optimization model is developed to obtain an optimal resource allocation schedule considering the operation cost, and the cost objective function proves to have convex behavior with optimal solutions. The performance of the operation system and its asymptotic behavior are derived with respect to different resource allocation schedules by formulating the operation processes. Finally, numerical experiments are conducted to verify the system’s performance and validity of the proposed model, and some insights are given about how to increase the terminal’s efficiency.

1. Introduction

Container terminals are crucial nodes in seaborne container transport supply chains. Since the world economy has rapidly developed, the throughput of container ports has increased. To keep a competitive position, terminal operators seek methods to handle the large number of containers with higher speed and lower cost. Automated container terminals have been used with the development of technology and automated equipment. They reduce labor cost and emissions while increasing the terminal efficiency. Containers in the automated container terminals are handled by quay cranes (QCs), AGVs and yard cranes (YCs) consecutively. Therefore, the schedules of each equipment and the number of equipment types used during operations are critical to terminal efficiency and cost.

Resource allocation is tasked with determining how many pieces of handling equipment are required to achieve operation efficiency while decreasing operating cost. Adding more equipment can increase the terminal efficiency but it may also lead to a lower utilization of equipment if the number of equipment pieces is too many. Meanwhile, the terminal productivity cannot be significantly increased by increasing the number of equipment pieces, and the congestion at the quayside and yard blocks does not increase in proportion to the amount of equipment in use. Moreover, there are a variety of types of equipment in automated container terminals, and the number of each type should be balance. For example, if too many AGVs are used, the efficiency of the QCs and YCs cannot corporate with that of the AGVs, which will lead to the congestion of AGVs at the quay side and yard blocks. Therefore, determination of the proper resource allocation requires a trade-off between the cost required to maintain the productivity and the handling speed necessary to complete the operation. From a system point of view, consideration of the congestion at each operation node and its incorporation into a formulation to determine the optimal resource allocation schedule is important in the design of the terminal operation system. This paper optimizes the resource allocation based on double cycling operations. Double cycling integrates unloading and loading operations on the same ship, the system’s operational processes being relatively complex. The results from Goodchild and Daganzo [1] have shown the benefits and effects of this technique on container terminal operations. Since it has changed the handling processes of container terminals, some decision problems in designing the double cycling operation systems need to be addressed. The resource allocation problem based on double cycling is challenging but critical to terminal management.

This paper develops an optimization model to determine the resource allocation schedule during handling operations and aims to identifying the proper amount of equipment needed to lower operating costs while ensuring handling efficiency. To model the congestion of AGVs at the quayside and yard blocks, a cyclic queuing network is applied from the AGVs’ perspective. We study the operating cost as a function of fleet size and prove its convex behavior if the number of QCs and YCs is constant. Additionally, the asymptotic behavior of performance indicators is derived, which provides reference for designing the terminal operation system with high efficiency and productivity. Numerical experiments are conducted to verify the system’s performance and validity of the proposed model.

The reminder of this paper is organized as follows. Section 2 presents a literature review on related research, and Section 3 models the handling system using a cyclic queuing network and derives selected properties, including asymptotic behavior and product form expressions. Section 4 proposes an optimization model to determine the proper amount of equipment and achieve the goal of minimizing operating cost. Section 5 discusses a case study on the performance of the double cycling operation system. Section 6 offers the conclusions of this paper.

2. Literature Review

This paper investigates the resource allocation problems of double cycling. The terminal operations are modeled by a closed queuing network and system performance of double cycling is analyzed. The most related literature to this paper is the related research on double cycling in container terminals and analyzing container terminal operations by simulation and queuing network.

Double cycling is a technique implemented to increase QC efficiency, and its effect on container terminals and scheduling problems have been researched. Goodchild and Daganzo [1] compared the benefit of this approach with that of single cycling on QCs. The effect on other operations in container terminals can also be found in Goodchild and Daganzo [2], who evaluated the changes in truck routes and estimated the expected truck number required to ensure terminal efficiency. The results showed that with double cycling, the cycling number of the operation decreased by 20%, and the handling time decreased by 10%. Other research on double cycling optimized QC scheduling to achieve the most benefit. For example, Zhang et al. [3] considered the impact of double cycling of quay cranes on yard crane operations and optimized the sequence of quay cranes loading and unloading to reduce the number of container turnovers at the yard blocks. Lee et al. [4] treated the double cycling of QCs as a flow shop problem and used the Sidney algorithm to solve the QC’s scheduling problem.

Analyzing the terminal operations systems is essential before making resource allocation schedules, optimizing task assignment, etc. There have been some simulation approaches to optimize the container terminal operations. Simulation provides a good result compared with mathematic programming. For example, Speer and Fischer [5] developed a simulation model to solve the yard crane scheduling problem for four different yard crane systems in automated container terminals. Vis et al. [6] studied the dispatch problems of trucks and the required number of trucks is minimized based on the time-window constraints. Simulation is used to validate the estimates by the analytical model. The simulation method is the most direct method in designing the operation system because it considers possible and realistic situations during handling operations. Liu et al. [7] uses simulation to analyze the impact of the two yard layouts of automated container terminals, and the number of vehicles was also determined for each scenario. Petering [8] proposed a discrete event simulation model to evaluate the terminal performance with different sizes of yard blocks. Petering et al. [9] propose a real-time yard crane control system to analyze different dispatching strategies. Sauri et al. [10] provided several simulation models considering cost to determine the optimal automated horizontal transport compared with a semi-automated system.

Another approach in quantifying the performance of container terminals and determining the required number of resources is queuing theory. Queuing theories have found wide application in the analysis of system performance and modeling of operational processes. Kang et al. [11] used a queuing network to model the operational system with the goal of optimizing the fleet size. The objective was to minimize the operating cost of the system, and a Markovian decision model was used to manage the real-time fleet size. Zhang et al. [12] used a closed queuing network to model terminal operations to study the stability of the system with higher variation of the handling time considering double cycling on the container terminals. Roy et al. [13] used a multi-class closed queuing network model to analyze both order picking and replenishment processes in a mobile fulfillment system storage zone. George and Xia [14] modeled the service network of a rental system as a closed queuing network and analyzed its service availability at the steady state, and the fleet size was subsequently optimized to maximize system profit. Hu and Liu [15] considered road congestion constraints to jointly optimize the parking capacities and fleet size in a car-sharing system. Mishra et al. [16] used a semi-open queuing network model for the interterminal transportation problem. Bounds for throughput time are estimated. Roy et al. [17] proposed a semi-open queuing network to compare the efficiency of AGV and ALV handling system. Roy et al. [18] estimated the throughput in automated container terminals by stochastic queuing models. Xiang X. and Liu C. [19] used a semi-open queuing network to estimate the performance of an automated container terminal. According to the above literature, queuing models provide a direct and efficient approach to design an operation system by analyzing its performance in terms of queuing length, service ability, equipment utilization, and operating cost, among other factors.

The above literature models the manufacturing systems by different queuing networks, for example, closed queuing network, semi-open network, etc. Exponential and general distributions are both used to analyze the system’s performance. However, different research problems have their own characteristics and should be modeled by different theories, and the solving methods are also different according to the specific queuing networks. This paper focuses on the resource allocation problem based on QC double cycling in container terminals. A trade-off is considered between the cost and the handling speed in determining the proper number of resources. Our contribution to modeling the resource allocation problem based on double cycling is twofold. First, this paper models the double cycling operations in container terminals by a cyclic queuing network to show its effect on terminal operations. The asymptotic behavior of the system is derived to provide some insights for designing the container terminal systems. The properties of the system show how to increase the overall terminal efficiency instead of simply optimizing one sub-operation. Second, a cost function is used to optimize the resource allocation schedules according to the system performance. Congestion at each station and the relationships among different operations are considered. It gives an insight for the terminal operators considering a trade-off of cost and efficiency to achieve specific goals, for example, save cost and make full use of equipment, etc.

3. Modeling the Handling Operations in Container Terminals

3.1. Problem Description

The flow of containers in the terminal contains handling operations at the quayside by QCs, transportation operations by AGVs and handling operations at the yard blocks by YCs. Each node contributes to the total terminal handling efficiency, and any delay affects the total handling time. AGVs carry an inbound container from the quay side and transports it to the inbound yard blocks, and then travel empty to the outbound yard blocks to carry an outbound container to the quayside. As shown in Figure 1, the overall operations include: (1) loaded by a QC with an import container; (2) carry the import container from the quayside to inbound yard; (3) unloaded by a YC; (4) empty travel from the inbound yard to outbound yard; (5) loaded by a YC with an export container; (6) carry the export container from the outbound yard to the quayside. With traditional operations, AGVs either go to inbound yard blocks or outbound yard blocks. We take unloading ship as an example, and the overall operations include: (1) loaded by a QC with an import container; (2) carry the import container from the quayside to inbound yard; (3) unloaded by a YC; (4) travel from the inbound yard to the quayside. The overall operations of the AGVs with double cycling increase the two stages compared to that of traditional operations, and the handling system is more complex. Research on the resource allocation is necessary to take better the advantage of double cycling.

Productivity and cost are often used to evaluate the terminal operations. Productivity can be increased by adding the number of handling equipment types during operations. Thereby the type of equipment and the number of equipment pieces are the two important decision problems. For example, although the AGV cost is the lowest, the cost of adding redundant AGVs might be negative because congestion can occur at the quayside and yard blocks, which lowers the turn-around efficiency of AGVs. Additionally, if only increasing the number of QCs, the workload at the yard blocks may be heavy. It is intuitive that the workload at the quayside and yard blocks should remain at an equilibrium state. An unbalanced operation rate results in a surplus of equipment utilization. Therefore, this paper optimizes the number of each type of equipment piece and analyzes the efficiency from the system’s view.

Since AGVs are responsible for transporting containers between the seaside and the yard blocks, the handling time at the seaside and yard blocks and the routes among them comprise a network system [20]. This network can also describe the overall flow of containers in the terminal, and all of the handling operations in a terminal are included in the system. Therefore, this terminal operation system can be modeled by a cyclic queuing network from the AGV perspective. A cyclic queuing network is a special case of closed queuing network, where the number of AGVs is constant. In the cyclic queuing network, the route of AGVs is cyclic, where they travel from the quayside to the inbound yard block, then to the outbound yard block and finally to the quayside. In this way, the AGVs are viewed as customers, and the quayside and yard blocks are viewed as server stations. The travel time between the quayside and yard blocks is also viewed as a virtual service station with infinite servers. The expected virtual service time is equal to the expected actual travel time. Therefore, with double cycling, the network system consists of six stations and K AGVs, as shown in Figure 2.

We make some assumptions about the closed queuing network, but they will not significantly affect the objectives of this paper. The assumptions are as follows:

• Specific information of the vessels and task assignments is not considered, because this paper provides an estimation of the system performance from the long-time run;
• The service times at all stations are assumed to follow exponential distributions. Exponential distribution is used, and we ignore the impact of it on the results because we mainly focus on the analysis of double cycling. The results of Koenigsberg and Lam [21] show there is small difference between the exponential and normal distributions;
• All servers have an FCFS (first come first service) policy;
• The buffer size of each queue is unlimited;
• All the AGVs complete an import and export container in a cycle;
• Road congestion is not considered in this paper.

The decision variables used to determine the optimal resource allocation schedule are described as follows:

K: Number of AGVs in the system;

$n_1$: Number of QCs;

$n_2$: Number of YCs in the inbound yard blocks;

$n_3$: Number of YCs in the outbound yard blocks;

The parameters of this model are described as follows:

$T$: Maximum average handling time of a ship;

$N$: Number of inbound or outbound containers (the number of inbound containers is equal to the number of outbound containers with double cycling);

$N_1$: Available number of QCs;

$N_2$: Available number of YCs in the inbound yard blocks;

$N_3$: Available number of YCs in the outbound yard blocks;

${\mu }_1$: Average service rate of QCs per minute;

${\mu }_2$: Average service rate of YCs in the inbound yard blocks per minute;

${\mu }_3$: Average service rate of YCs in the outbound yard blocks per minute;

${\mu }_4$: Average service rate from the quayside to the inbound blocks, whose reciprocal is the travel time of the AGVs from the quayside to the inbound blocks;

${\mu }_5$: Average service rate from the inbound blocks to the outbound blocks, whose reciprocal is the travel time of the AGVs from the inbound blocks to the outbound blocks;

${\mu }_6$: Average service rate from the outbound blocks to the quayside, whose reciprocal is the travel time of the AGVs from the outbound blocks to the quayside.

3.2. Product Form Equilibrium

This cyclic queuing network is represented using continuous-time Markov chains with discrete state spaces. Let $k_1,k_2,k_3,k_4,k_5,k_6$ represent the steady state of the system, corresponding to the number of AGVs at the quayside, at the inbound yard blocks, at the outbound yard blocks, on the way from the quayside to the inbound yard block, on the way from the inbound yard blocks to the outbound yard blocks and on the way from the outbound yard blocks to the quayside, respectively. The sum of the AGVs at each stage is equal to K, that is $k_1+k_2+k_3+k_4+k_5+k_6=K$. The cyclic queuing system at the steady state has product form expressions. Gordon and Newell [22] give the steady probability of each state as follows:

$$P(k_1,k_2,k_3,k_4,k_5,k_6)={\displaystyle \prod }_{i=1}^6\frac{{(X_i)}^{k_i}}{l_i}G{(K,n_1,n_2,n_3)}^{-1}$$

(1)

$$\mathrm{where} l_j=\{\begin{array}{c}{n}_j!n_j^{k_j-n_j},k_j\ge n_j\\ k_j!,k_j\le n_j\end{array},X_i=\frac{{\mu }_1}{{\mu }_i}$$

(2)

$${\displaystyle \sum _{\Theta }P\left(k_1,k_2,k_3,k_4,k_5,k_6\right)}=1$$

(3)

In addition, $\Theta$ is the set of system states, and the sum of all probabilities is equal to 1.

$G\left(K,n_1,n_2,n_3\right)$ is the normalization constant used to let the sum of the probabilities be equal to 1, and it is a function of the number of each type of equipment:

$$G\left(K,n_1,n_2,n_3\right)={\displaystyle \sum _{\Theta }\left({\displaystyle \prod }_{i=1}^6\frac{{\left(X_i\right)}^{k_i}}{l_i}\right)}$$

(4)

The marginal distribution of the queuing length at station i (there are k AGVs at station (i)) can be obtained as follows:

$$P\left(k_i=k\right)=\frac{{\left(X_i\right)}^k}{l_i}\cdot \left({\displaystyle \sum _{{\displaystyle \sum _{j\ne i}{k}_j=K-k}}{\displaystyle \prod }_{j=1,j\ne i}^6\frac{{\left(X_j\right)}^{k_j}}{l_j}}\right)\cdot G{\left(K,n_1,n_2,n_3\right)}^{-1}=\frac{{\left(X_i\right)}^k\cdot G_i\left(K-k,n_1,n_2,n_3\right)}{l_i\cdot G\left(K,n_1,n_2,n_3\right)}$$

(5)

where $G_i\left(K-k,n_1,n_2,n_3\right)$ represents the normalization constant of a cyclic queuing network that the population is $K-k$ and the service stations except for i are included.

3.3. Performance Analysis

Let $I_i\left(K,n_1,n_2,n_3\right)$ represent the probability that the servers in the i-th stage are idle. Thus, $I_i\left(K,n_1,n_2,n_3\right)$ equals the probability that the number of AGVs in the i-th stage is zero, and the derivation is given in Appendix A:

$$I_i\left(K,n_1,n_2,n_3\right)=P\left(k_i=0\right)=\frac{G\left(K,n_1,n_2,n_3\right)-X_i\cdot G\left(K-1,n_1,n_2,n_3\right)}{G\left(K,n_1,n_2,n_3\right)}$$

(6)

The utilization in each stage is the percentage probability that there is always at least one AGV at each stage. From a mathematical perspective, this value equals the probability that the servers in the i-th stage are busy at the steady state. Let $B_i\left(K,n_1,n_2,n_3\right)$ represent the probability that the servers in the i-th stage are busy with total K AGVs. Thus, the sum of $B_i\left(K,n_1,n_2,n_3\right)$ and $I_i\left(K,n_1,n_2,n_3\right)$ is equal to 1:

$$B_i\left(K,n_1,n_2,n_3\right)=1-I_i\left(K,n_1,n_2,n_3\right)=\frac{X_i\cdot G\left(K-1,n_1,n_2,n_3\right)}{G\left(K,n_1,n_2,n_3\right)}$$

(7)

Theorem 1.

The utilization of equipment at the quayside and yard blocks is constrained by the stage with the minimum service rate.

Proof of Theorem 1.

Let $X_{\alpha }$ denote the maximum relative service ratio among $X_i\left(i=1,2,3\right)$, which is also the stage with the minimum service rate. According to Equation (7), it is straightforward to obtain the following with a non-negative value of $G\left(K,n_1,n_2,n_3\right)$:

$$B_i\left(K,n_1,n_2,n_3\right)<B_{\alpha }\left(K,n_1,n_2,n_3\right)\le 1,i=1,2,3$$

(8)

If the number of AGVs K continues to increase, the stage with the minimum service rate is finally saturated, and the utilization of equipment approaches to 1:

$$\underset{K\to \infty }{\lim }{B}_{\alpha }\left(K,n_1,n_2,n_3\right)=1$$

(9)

According to Equation (7), the following can be derived:

$$\frac{B_i\left(K,n_1,n_2,n_3\right)}{B_{\alpha }\left(K,n_1,n_2,n_3\right)}=\frac{X_i}{X_{\alpha }},i=1,2,3$$

(10)

Equation (9) gives the limit value of utilization with the minimum service rate, and when $K\to \infty$, the probability that the servers in the i-th stage are busy has the following approach:

$$\underset{K\to \infty }{\lim }{B}_i\left(K,n_1,n_2,n_3\right)=\underset{K\to \infty }{\lim }\frac{B_i\left(K,n_1,n_2,n_3\right)}{B_{\alpha }\left(K,n_1,n_2,n_3\right)}=\frac{X_i}{X_{\alpha }},i=1,2,3$$

(11)

From Equation (11), the utilization of the equipment at each stage is constrained by the stage with the minimum service rate, as described in Theorem 1. This statement suggests that the terminals should increase the service ability of the minimum service rate to reach a higher efficiency for the overall operation.

The actual throughput of a stage is the expected number of departural AGVs from each stage per unit time. Let $T{R}_i\left(K,n_1,n_2,n_3\right)$ represent the actual throughput of the i-th stage, and its expression is given as follows:

$$T{R}_i\left(K,n_1,n_2,n_3\right)=B_i\left(K,n_1,n_2,n_3\right)\cdot m_i\cdot {\mu }_i$$

(12)

 □

Corollary 1.

The utilization of equipment at the quayside and yard blocks is a non-decreasing concave function.

Proof of Corollary 1.

The throughput at each stage $T{R}_i\left(K,n_1,n_2,n_3\right)$ is a non-decreasing concave function of the total AGV number K. The proof can be found in the work of Shantkikumar and Yao [23]. According to Equation (12), the utilization can be represented as $B_i\left(K,n_1,n_2,n_3\right)=\frac{T{R}_i\left(K,n_1,n_2,n_3\right)}{m_i\cdot {\mu }_i}$. The parameters $m_i,{\mu }_i$ are constant values; therefore, $B_i\left(K,n_1,n_2,n_3\right)$ and $T{R}_i\left(K,n_1,n_2,n_3\right)$ have the same properties. □

Definition 1 (Bottleneck operation).

The bottleneck operation in the steady state of the operation system is defined as the busiest stage in which the utilization of equipment approaches 1 when the number of AGVs tends toward infinity.

Proposition 1.

Only increasing the service ability in the bottleneck stations can enhance the overall operation efficiency.

Proof of Proposition 1.

Theorem 1 implies that the bottleneck station in the flow of containers in the terminal results from the maximum service time. Therefore, Proposition 1 can easily be obtained. □

The handling ability is the number of AGVs that are serviced per unit time at each stage. We use it to represent the handling ability of the i-th stage. The detailed derivation is shown in Appendix B:

$$P_i\left(K,n_1,n_2,n_3\right)=\{\begin{array}{l}{\displaystyle \sum _{k=1}^{m_i-1}P\left(k_i=k\right)\cdot k\cdot {\mu }_i}+{\displaystyle \sum _{k=m_i}^{}P\left(k_i=k\right)\cdot m_i\cdot {\mu }_i},i=1,2,3\\ \frac{{\mu }_i}{G\left(K,n_1,n_2,n_3\right)}{\displaystyle \sum _{j=1}^K{\left(X_i\right)}^j\cdot G\left(K-j,n_1,n_2,n_3\right)},i=4,5,6\end{array}$$

(13)

The expected number of AGVs in the i-th stage can be represented as follows:

$$L_i\left(K,n_1,n_2,n_3\right)={\displaystyle \sum _{\Theta }k\cdot }P\left(k_i=k\right)=\frac{{\displaystyle \sum _{j=1}^{K}j\cdot {\left(X_i\right)}^j\cdot G_i\left(K-j,n_1,n_2,n_3\right)}}{l_i\cdot G\left(K,n_1,n_2,n_3\right)}$$

(14)

Little’s formula can also be used in the cyclic queuing system [24]. The average turn-around time of AGVs in the i-th stage ${\mu }_i,\left(i=1,2,3\right)$ can be obtained from $W_i\left(K,n_1,n_2,n_3\right)=\frac{L_i\left(K,n_1,n_2,n_3\right)}{m_i\cdot {\mu }_i}$. The turn-around time on the travel route $\left(i=4,5,6\right)$ is the actual service time and is equal to ${\mu }_i,\left(i=4,5,6\right)$:

$$W_i\left(K,n_1,n_2,n_3\right)=\{\begin{array}{c}\frac{{\displaystyle \sum _{j=1}^{K}j\cdot {\left(X_i\right)}^j\cdot G_i\left(K-j,n_1,n_2,n_3\right)}}{m_i\cdot {\mu }_i\cdot l_i\cdot G\left(K,n_1,n_2,n_3\right)},i=1,2,3\\ {\mu }_i,i=4,5,6\end{array}$$

(15)

The waiting time of AGVs in the i-th stage equals to the turn-around time minus the service time:

$$w_i\left(K,n_1,n_2,n_3\right)=W_i\left(K,n_1,n_2,n_3\right)-{\mu }_i,i=1,2,3,4,5,6$$

(16)

The throughput of the system is the number of containers handled per hour. This value can be treated as the number of arriving AGVs per unit time at each station, which is the reciprocal of the visit frequency to each station. Because an AGV carries two containers (an outbound and an inbound container) in a cycle, the real throughput is doubled.

$$TH\left(K,n_1,n_2,n_3\right)=\frac{120K}{{\displaystyle \sum _i{W}_i\left(K,n_1,n_2,n_3\right)}}$$

(17)

Lemma 1.

The throughput of the system $TH\left(K,n_1,n_2,n_3\right)$ is a non-decreasing concave function of the total AGV number K. The proof can be found in Shantkikumar and Yao [23]. Besides, the throughput of the system is equal to the lowest throughput among the stages, $TH\left(K,n_1,n_2,n_3\right)=T{R}_{\beta }\left(K,n_1,n_2,n_3\right)$, and $T{R}_{\beta }\left(K,n_1,n_2,n_3\right)$ means the lowest value among $T{R}_i\left(K,n_1,n_2,n_3\right)$.

4. Resource Allocation Optimization

4.1. Resource Allocation Model

Let $C_t,C_q and C_{yc}$ represent the costs of AGVs, QCs and YCs per unit time, respectively. To complete the handling tasks, the total cost of operations equals the number of equipment types multiplied by the cost per unit time multiplied by the total operation time. The resource allocation optimization model is as follows:

Obj.

$$\min F_1\left(K,n_1,n_2,n_3\right)=\left(C_t\cdot K+C_q\cdot n_1+C_{yc}\left(n_2+n_3\right)\right)\cdot \frac{{\displaystyle \sum _i{W}_i\left(K,n_1,n_2,n_3\right)}}{K}\cdot N$$

(18)

S.t.

$$\frac{{\displaystyle \sum _i{W}_i\left(K,n_1,n_2,n_3\right)}}{60K}\cdot N\le T$$

(19)

$$n_1\le N_1$$

(20)

$$n_2\le N_2$$

(21)

$$n_3\le N_3$$

(22)

Equation (18) is the objective function used to minimize the operating cost of the system. Constraint (19) is the time constraint stating that the operation time should not exceed the maximum available handling time, and Constraints (20)–(22) are the resource constraints stating that the number of each type of equipment should not exceed the maximum available resource.

Theorem 2.

The total cost function $F_1\left(K,n_1,n_2,n_3\right)$ is a convex function of the total number of AGVs K.

Proof of Theorem 2.

$TH\left(K,n_1,n_2,n_3\right)$ is the actual throughput of the system, which is also a concave function of K. Therefore, the time that the system requires to handle all containers can be expressed as $\frac{N}{TH\left(K,n_1,n_2,n_3\right)}$. The objective function can be replaced by $\min F_1\left(K\right)=\left(C_t\cdot K+V\right)\cdot \frac{N}{TH\left(K,n_1,n_2,n_3\right)}$ as a simplified version, where $C_q\cdot n_1+C_{yc}\left(n_2+n_3\right)=V$ is a constant value. According to the above analysis, the AGV population K and $\frac{1}{TH\left(K,n_1,n_2,n_3\right)}$ are both convex functions; therefore, the product of the two items is also a convex function [25]. □

Corollary 2.

The optimal solution of the first objective function exists and has at most two optimal values with the given number of QCs and YCs.

Proof of Corollary  2.

The first derivation of the first objective function is shown as follows:

$$\frac{\partial F_1\left(K,n_1,n_2,n_3\right)}{\partial K}=\left(C_t\cdot TH\left(K,n_1,n_2,n_3\right)-\left(C_t\cdot K+V\right)\cdot \frac{\partial TH\left(K,n_1,n_2,n_3\right)}{\partial K}\right)\cdot \frac{N}{TH\left(K,n_1,n_2,n_3\right)}$$

(23)

Since $TH\left(K,n_1,n_2,n_3\right)$ is a concave function of K, $\frac{\partial TH\left(K,n_1,n_2,n_3\right)}{\partial K}$ decreases, and $\underset{K\to \infty }{\lim }\frac{\partial TH\left(K,n_1,n_2,n_3\right)}{\partial K}=0$. It is intuitive that if only a few AGVs exist, the marginal throughput of the system increases quickly, and $\underset{K\to 0}{\lim }\frac{\partial TH\left(K,n_1,n_2,n_3\right)}{\partial K}=\infty$. Therefore, $\underset{K\to 0}{\lim }\frac{\partial F\left(K,n_1,n_2,n_3\right)}{\partial K}<0$, and $\underset{K\to \infty }{\lim }\frac{\partial F_1\left(K,n_1,n_2,n_3\right)}{\partial K}>0$. There exists a value of $\xi$ that guarantees that $\frac{\partial F_1\left(K,n_1,n_2,n_3\right)}{\partial K}{|}_{K=\xi }=0$. Therefore, $\xi$ is the optimal number of AGVs, but if $\xi$ is not integral, two integral optimal values can be obtained. □

4.2. Performance Calculation

This paper uses the convolution algorithm proposed by Buzen [24] to compute the normalization constant $G\left(K\right)$ of the equilibrium distribution expressions. Equation (4) showed that $G\left(K\right)$ is the summation of $(\begin{array}{c}N+K+1\\ K\end{array})$ terms, which makes it complex to approximate. Buzen’s algorithm [24] uses an iterative algorithm with which the values of $G\left(1\right),G\left(2\right),\dots ,G\left(K\right)$ can be computed using a total of $N\cdot K$ multiplications and $N\cdot K$ additions. This algorithm consists of the following functions and relationships:

$$g(k,n)={\displaystyle \sum _{(n,k)}({\displaystyle \prod }_{i=1}^n\frac{{(X_i)}^{k_i}}{l_i})}$$

(24)

where n is the number of server stations and k is the number of customers in the system. Therefore,

$$G(k)=g(k,N),\forall k=0,1,\dots ,K$$

(25)

$$g(k,n)={\displaystyle \sum _{k_i=0}({\displaystyle \prod }_{i=1}^n\frac{{(X_i)}^{k_i}}{l_i})}+{\displaystyle \sum _{k_i\ne 0}({\displaystyle \prod }_{i=1}^n\frac{{(X_i)}^{k_i}}{l_i})}=g(k,n-1)+X_n\cdot g(k-1,n)$$

(26)

The following can be derived from the above iterative equations:

$$g(k,1)={(X_1)}^n,\forall k=0,1,\dots ,K$$

(27)

$$g(0,n)=1,\forall n=0,1,\dots ,N$$

(28)

The values $G(1),G(2),\dots ,G(K)$ can be obtained with the above iterative relationships and equations. To solve Equation (19), the performance indicators of each resource allocation schedule should be computed iteratively before computing the cost function value.

5. Numerical Experiments

This section presents a case study to show the validity of the proposed model and algorithm. By changing the number of equipment, this paper analyzes the system performance with different resource allocations. For example, 1QC + 1YC + 1YC means 1 QC is equipped with 1 YC at the outbound yard blocks and 1 YC at the inbound yard blocks. The resource allocation is not for only one ship but for multiple ships, because the assumption in this paper states that specific information of ships is not considered. The parameter values are obtained from an investigation at Tianjin Port, which are shown in

Table 1. Parameter values.

Parameters	Value	Parameters	Value
Expected service rate of QCs (moves per minute)	0.35	Expected travel time from inbound yard blocks to outbound yard blocks (min)	5
Expected travel time from quayside to inbound yard blocks (min)	3.57	Expected service rate of YCs at outbound yard blocks (moves per minute)	0.49
Expected service rate of YCs at inbound yard blocks (moves per minute)	0.41	Expected travel time from quayside to outbound yard blocks (min)	3.45
Total containers (TEU)	4000	Cost of QCs per unit time	4.5
Cost of YCs per unit time	1.75	Cost of AGVs per unit time	1

Source: Investigation at Tianjin Port.

. The cost of equipment per unit time contains the labor cost, power cost, maintenance cost and depreciation cost. Based on the investigation results and the proportion from Kang et al. [11], the proportion of the cost of QCs, YCs and AGVs per unit time is revised as 4.5:1.75:1. Other parameters are shown in Table 1 as follows:

5.1. Convex Behavior of the Objective Function

When the number of QCs and YCs is constant, Section 4 gives the convex behavior of the objective function as the number of AGV changes. The handling cost has a minimum value for each resource allocation schedule of QCs and YCs. The evaluation results are shown in Figure 3 and

Table 2. Cost analysis with different resource allocation schedules.

Resource Allocation Schedules	Number of AGVs
	3	4	5	6	7	8	9	10	11	12	13	14	15	16
1QC + 1YC + 1YC	3520	3149	3015	3000	3050	3141	3256	3389	3534	3688	3848	4014	4183	4355
1QC + 2YC + 2YC	4473	3852	3602	3546	3600	3718	3872	4044	4226	4413	4602	4792	4982	5173
2QC + 2YC + 2YC	5484	4368	3730	3341	3098	2950	2866	2827	2820	2837	2871	2918	2974	3038
2QC + 3YC + 3YC	6503	5119	4313	3806	3479	3270	3145	3082	3066	3084	3128	3189	3262	3344
3QC + 3YC + 3YC	7773	6046	5015	4335	3861	3519	3268	3084	2951	2857	2794	2756	2738	2736

. Once the available QCs and YCs are determined, the optimal number of AGVs can be obtained according to the results. For example, the resource allocation of 1QC + 1YC + 1YC and 1QC + 2YC + 2YC has the lowest cost when the number of AGVs is 6. In addition, the terminal can also reach a same handling cost by changing the number of QCs and YCs. For example, the operating cost of 1QC + 2YC + 2YC with 5 AGVs is almost equal to that of 3QC + 3YC + 3YC with 8 AGVs. However, although different resource allocation schedules can obtain the same handling cost, other system performance indicators of each resource allocation schedules are different. This observation is intuitive because the operating cost is affected by the amount of equipment and the total operation time. The additional number of equipment resources results in a shorter handling time, but the equipment cost per unit time is higher.

5.2. Validity of the Proposed Model

Section 3 and Section 4 analyze the system performance of container terminals with double cycling. This section verifies the validity of the proposed model with different resource allocation schedules.

5.2.1. Operation Time

Total operation time of a container is a key criterion used to evaluate the handling ability of a container terminal. Terminal operators strive to decrease the handling time of a ship so that more ships can be serviced in a busy period. As shown in Figure 4 and

Table 3. Total handling time analysis with different resource allocation schedules (hours).

Resource Allocation Schedules	Number of AGVs
	3	4	5	6	7	8	9	10	11	12	13	14	15	16
1QC + 1YC + 1YC	317	260	230	213	202	195	190	187	185	183	182	182	181	180
1QC + 2YC + 2YC	304	245	216	200	193	189	187	186	186	186	186	186	187	187
2QC + 2YC + 2YC	286	216	176	150	134	122	114	108	104	101	98	97	95	94
2QC + 3YC + 3YC	285	215	174	148	130	118	109	103	100	97	95	94	94	93
3QC + 3YC + 3YC	285	214	171	143	123	109	98	90	84	79	75	72	70	67

, the total operation time decreases by increasing the number of QCs, YCs and AGVs while the decreasing rate decreases as the number of AGVs increases. For example, with the resource allocation of 1QC + 1YC + 1YC and 1QC + 2YC + 2YC, when the number of AGVs exceeds eight, the decreasing rate in the operation time rate becomes slower. It will cost the terminal more to reduce the handling time by one hour by adding more AGVs. The marginal cost becomes higher, and the benefit is little. Therefore, it is necessary for the terminal operators to determine the amount of equipment necessary to increase profit, and a trade-off should be made between the operation time and operating cost. Additionally, the difference between the handling time of resource allocation 1QC + 1YC + 1YC and 1QC + 2YC + 2YC is small, especially when the number of AGVs is large, because the handling operations are constrained by the service rate of QCs and YCs.

5.2.2. Idle Rate of QCs and YCs

Generally, the idling cost of a QC is more expensive than that of other equipment, and the efficiency of QCs directly affects the turn-around time of the vessels. Therefore, terminal operators prefer to increase the number of YCs and AGVs to ensure QCs’ utilization. As shown in Figure 5, by increasing the number of YCs and AGVs, idle rate of QCs decreases. With the resource allocation of 1QC + 1YC + 1YC, 14 AGVs make the fullest use of the QCs. It can be inferred that to make full use of the QCs with the resource allocation of 3QC + 3YC + 3YC, approximately 42 AGVs are needed, which adds significant cost. Comparing the results in Figure 6 vertically, adding YCs can also produce a decrease in the idle rate of the QC, and the cost per YC is much more expensive than the AGV cost. Therefore, the idle rate of the QC is a reference for the container terminal operators to allocate equipment together with other indicators, such as operating cost and handling efficiency.

5.2.3. Waiting Time of AGVs

AGVs are responsible for the horizontal transportation of containers in the terminal and must wait for containers if there are no available QCs and YCs when arriving at the quayside and yard blocks. The waiting time has an important effect on terminal congestion and energy consumption. We estimate the average waiting time of AGVs, which is shown in Figure 6. However, although more QCs and YCs lead to little energy consumption, the cost of QCs and YCs is even more expensive. Therefore, terminal operators should equip proper numbers of AGVs to balance cost and energy consumption.

5.3. Sensitive Analysis

To test the impact of the unit cost of QCs, YCs and AGVs on the total cost, we conduct a sensitive analysis. We set four scenarios as follows: (1) the unit cost of QCs, YCs and AGVs is set to 3.5, 1.5, 1.25 respectively, which changes the unit cost of QCs, YCs and AGVs; (2) the unit cost of QCs, YCs and AGVs is set to 4, 1.75, 1.5 respectively, which increases the unit cost of the AGVs; (3) the unit cost of QCs, YCs and AGVs is set to 3.5, 1.75, 1, respectively, which decreases the unit cost of QCs; (4) the unit cost of QCs, YCs and AGVs is set to 4, 2, 1, respectively, which increases the unit cost of the YCs. The results are shown in Figure 7. From the results, the changes of the unit cost of the QCs can decrease the total handling cost. The convex characteristics are not changed whether the unit cost changes.

5.4. Resource Allocation Insights

The total handling time makes the operations flexible when choosing the proper number of resources. During the busy period, the terminal operators can choose a resource allocation schedule with high productivity. Thus, more ships can berth at the ports, increasing the profit to the container terminals. In stagnant periods, the terminal operators can choose a resource allocation with lower operating cost to save money.

Theorem 1 gives suggestions for increasing the service ability of the bottleneck operations to enhance the overall operational efficiency. It is useless to change the service ability of other operations to gain an increase in the overall productivity. Terminal operations should first identify the bottleneck operations and increase the service rate at those operations. However, productivity cannot be improved until the service ability at each stage is balanced.

6. Conclusions

This paper addressed the resource allocation problem to determine the amount of equipment based on the QC double cycling. The processes of container flow in terminals were modeled using a closed queuing network, including operations at the quayside and at the yard blocks, and horizontal transportation. The congestion of AGVs at the quayside and yard blocks and the utilization of each type of equipment were considered in approximating the turn-around time of the AGVs. The objective of the proposed optimization model was to lower the operating cost while ensuring the overall handling efficiency. The operating cost was expressed as a function of the number of resources, which was shown to exhibit convex behavior if the number of QCs and YCs is given. In addition, the asymptotic behavior of the performance indicators reveals that the overall terminal efficiency is limited by the station with the lowest service rate. These analysis results offer insights to terminal operators for deciding how to increase the system efficiency and which types of equipment can be added to increase terminal productivity. Finally, numerical experiments were performed to illustrate the efficiency of the model and the validity of the system performance.

The handling equipment in this research primarily includes QCs, YCs and AGVs. As technology has developed in the terminals, automated lifted vehicles (ALVs) have already been used in the operation system instead of YCs and QCs to lift the containers. Therefore, QCs and YCs do not need to wait for ALVs, and containers can be stacked on the ground temporarily. We intend to further explore the efficiency of this new handling system in a future study.