Introduction

Maritime transportation plays an important role in international import and export commodity transportation. According to United Nations Conference on Trade and Development statistics, maritime transportation has become the main mode of transportation for international trade1. With the rapid development of global trade activities, greenhouse gas emissions generated by maritime transportation are increasing. The International Maritime Organization (IMO) predicted that global carbon dioxide emissions from shipping will increase from 10 million tons in 2018 to 10–15 million tons in 20502. In order to reduce carbon emissions from shipping industry, some shipping organizations and countries in the world have successively proposed carbon emission reduction strategies specially for shipping. For example, IMO Initial GHG Strategy clearly stated that based on 2008, the emission from the shipping will be reduced by 40% by 2030 and 70% by 20503. The port is a typical energy-intensive industry, accounting for 3% of global greenhouse gas emissions4. The construction of low-carbon ports has become a global consensus. Some countries have put forward clear requirements for the construction of low-carbon ports5. For example, the United States Environment Protection Agency has detailed regulations on standards such as air quality and micro-particles in ports. Low-carbon port is an environmentally friendly port concept, which requires that the emissions of all cargo transportation and handling activities in the port are very low. Container trucks transportation emits a large amount of NOx and CO2, which will not only pollute the port, but also reduce the urban air quality6, and is one of the important sources of emissions from ports7.

Container transportation has become one of the most common modes of transportation in international trade8. When containers are unloaded from the container vessel, they are transported by container trucks to the yard for temporary storage. Therefore, container trucks are important equipment to connect the quay crane and the yard9. In recent decades, the trend of large-scale container vessels has become increasingly evident10. Because large container vessels have strict requirements on the berthing time of the port, a large number of container trucks arrive at the port for loading and unloading at the same time during the peak period, which is easy to cause queuing, congestion, emissions and other problems in the terminal gate, yard and operation line11. Therefore, the container truck allocation problem has always been the focus of attention of the container terminal12. A reasonable container truck allocation scheme can not only improve the operation efficiency of container terminal, but also reduce transportation time and ship docking time, thereby reducing environmental pollution13. The import and export blocks are important parts of the container terminal or even the hub of the container truck transportation. Therefore, the key point of the port container truck routing optimization is the scheduling optimization between the import blocks and the export blocks. The container terminal’s “quay crane - container trucks - import and export block” operation optimization has become an important means to improve port efficiency.

In terms of such circumstance, after Literature View in Sect. 2, we conducted our study in the following order: firstly, in Sect. 3, we construct a two-level “quay crane - container trucks - import and export block” container truck routing optimization model to reduce the total cost and emission of container truck transportation in container terminals. Then secondly in Sect. 4, we designed an improved PSO algorithm to solve the model. At last, in Sect. 5, we take Zhifu Bay Port as a practical case to apply the model, and performed sensitivity analysis for its carbon tax and the number of container truck.

In this study, we endeavor to solve the following problems:

  1. (1)

    How to allocate the container truck task of container terminal in a “quay crane - container trucks - import and export block” mode?

  2. (2)

    What is the impact of emission policies on the optimization of container truck transportation routes?

  3. (3)

    How does the number of container trucks affect the optimization of container truck transportation routes?

Routing optimization of container trucks is a practical and complex problem that should be solved by ports. The essential contribution of this research is to offer an innovated approach and solution to this conundrum.

Literature review

As a complex system, the allocation and scheduling of various equipment resources directly affect the operation efficiency of the whole container terminal system. Many scholars have studied the optimization of container trucks transportation14. Caballini et al.15 showed that inappropriate container trucks deployment schemes will cause waste of terminal resources and also cause container trucks congestion problem. Yet some other scholars hold that the container trucks reservation strategy can reduce the waiting time in the port16,17.

Phan and Kim18 designed a new container truck reservation process. The container trucks and the terminal jointly determine the container trucks operation plan and arrival time. Through this process, the waiting time of the container trucks can be effectively constructed, which is convenient for the container trucks scheduling of the terminal. Rijal et al.19 believed that container truck routing optimization and reservation are the key factors for container terminal optimization, and used a simulation method to study those two problems. Sun et al.20 proposed a data-driven method to solve the problem of container truck reservation quota optimization, and analyzed the relationship between the number of container trucks arriving at the terminal and their total turnover time in the port. The optimization problem of container truck transportation is relatively complex, so many scholars used intelligent algorithms to solve the problems. Amini et al.21 studied the scheduling problem considering the random distribution of container trucks failures, established a bi-objective linear mathematical model with the shortest queue length and the highest system reliability, and used an improved genetic algorithm to solve the model. Shahmardan22 focused on the scheduling optimization problem of inbound and outbound trucks cross docking centers, constructed a mixed integer programming model with the decision goal of minimizing the trucks travel time, and proposed a hybrid heuristic simulated annealing algorithm to solve the model. Guo et al.23 tried to solve the scheduling optimization problem of multi-vehicle tractor and semitrailer in container port, and designed a two-stage heuristic algorithm to solve the container truck routing optimization model. Wang et al.24 constructed a bi-level programming model of tractor network transportation in port hinterland, and designed a two-stage hybrid heuristic algorithm to solve the problem. Some scholars have also conducted research on auxiliary equipment and information collection. For example, Yang et al.25 brought up an auxiliary wheel of tractor, which is equipped with a 360°steering auxiliary wheel in the front of the tractor to achieve efficient headland steering. The test results showed that compared with the traditional steering method, the auxiliary wheel steering method improves the time efficiency, driving distance and occupied space of the headland by 50%, 80% and 50% respectively. Xu et al.26 founded a new information collection method for unmanned tractor in automatic lane driving to optimize the driving route of unmanned tractor.

With the construction of port emission reduction, the emission problem of container trucks has also attracted the attention of some scholars27,28. For example, Yu et al.29 established the emission calculation model of the yard tractor based on the queuing theory. Their result showed that the emissions of the yard tractor are closely related to the location of the container. Wang et al.30 adopted a method for calculating the carbon dioxide emissions of port container inland distribution network, which first deployed the method of Intergovernmental Panel on Climate Change to calculate CO2 emissions from PCD, and then used the local indicators of spatial association cluster map to detect the spatial clusters and spatial outliers. Lastly, the spatial Durbin model was applied to identify the main driving factors. This paper verified the method by taking 30 container ports in China as examples. Eglynas et al.31 measured the instantaneous energy consumption of port diesel trucks under various conditions, and designed an improved model to reduce the energy consumption level of diesel trucks and also reduce carbon emissions. Guo et al.32 designed a method for estimating emissions based on inland container multimodal transport networks. Taking Shanghai Port and the hinterland of the Yangtze River Delta as examples, they estimated the emissions of container trucks in the container multimodal transport network over the past decade. Bai et al.33 was concentrated on the pollutant emission calculation method of liquid natural gas (LNG) tractor truck in port. Tang et al.34 compared the emission levels of container trucks with different energy types in the port, and took the analysis of Ningbo Zhoushan Port as an example to calculate. Container trucks congestion in ports is also closely related to emissions. Lee et al.35 discussed the problem of container truck congestion and cost optimization considering ship replenishment, and considered that container truck congestion will directly lead to an increase in emissions. Facchini et al.36 conceived a scheme of adding dry port outside the sea port, which can not only solve the problem of sea port container truck congestion, but also reduce the emissions of container trucks by 11%. Ding et al.37 wielded a two-stage integer-optimized model to optimize the container truck routing optimization problem in order to reduce emissions in container terminals, and used the North Operation Area in Shanghai Yangshan Deep-Water Port as a case to verify the validity of the model. Enabling the use of new energy in container trucks is also an effective means to reduce emissions. Lombardi et al.38 proved the effect of a robust energy management strategy for fuel cell/battery hybrid tractors, which can dynamically adjust the operation of fuel cell/battery to achieve minimum energy consumption.

The construction of low-carbon terminals has become an important direction for the future development of green ports. The optimization of container truck routes is of great significance for reducing emissions from terminal operations. The above literatures studied the reservation, congestion, and emission measurement of container trucks’ routing optimization, but they have rarely involved the import block and the export block, or the uncertainty of yard capacity and ship berthing time are not considered, either. The considering of simultaneous operation of multi-crane and of the ship docking time is necessary for container truck transportation optimization. How to optimize the container truck route among multiple quay cranes, import block and export block, considering the impact of emission policies, is a practical problem that must be solved in the construction of green ports.

This paper studied the container truck routing optimization problem of “quay crane - container trucks - import and export block” considering carbon emissions. Because the capacity of the import and export container area is easily affected by the size of the site and the height of the container stacking, this study sets the volume capacity of the import and export container and the capacity of the export container area as uncertain variables to be more in line with the actual situation. Carbon quota policy is the main policy adopted by governments to control carbon emissions39. Therefore, this paper also considers the government’s implementation of carbon quota policy for port truck transportation, that is, to give a certain carbon emission quota for truck transportation. If the quota exceeds the actual emission, the port can sell the excess part through the carbon trading market to obtain certain income. If the actual emission exceeds the quota, the port needs to pay carbon tax to the government for the excess part. This paper constructs a two-level optimization model: Level I is container truck routing optimization model with the lowest total transportation cost as the goal, and each container truck transportation is regarded as a job allocation. Level II is container truck no-load routing optimization model with the minimum job allocation cost as the goal.

Model construction

Problem description

When a container vessel docks at a terminal, there are usually multiple quay cranes loading and unloading containers at the same time. The unloaded containers are transported from the terminal to the import block, and the containers that need to be loaded are from the export block. The operation process of the container trucks is shown in Fig. 1. In order to meet the container vessel’s loading and unloading requirements, there are three ways of container truck transportation, the first is quay crane area → import block → export block → quay crane area, and this way can meet the container vessel’s loading and unloading requirements at the same time, recorded as A1; the second is quay crane area → import block → quay crane area, which only meets the needs of container vessel’s unloading requirement, recorded as A2; the third is quay crane area → export block → quay crane area, and this approach only meets the container vessel’s packing requirement, recorded as A3. In order to promote the development of green ports, the government imposes carbon emission restrictions on container truck transportation. Port companies can sell carbon emission rights that do not meet carbon emission restrictions. If the carbon emission rights exceed the limit, they need to buy from the carbon trading market. The route of container trucks is shown in Fig. 1.

Figure 1
figure 1

Container trucks’ “quay crane - container trucks - import and export block” transportation route.

Full size image

The fuel consumptions of heavy load and no-load container trucks are different, so the fuel cost and emissions are also different. In the A1 route, the level of quay crane area → import block and export block → quay crane area is the heavy load level of the container truck, and the level of import block → export block is the no-load level. In the A2 route, quay crane area → import block stage is the heavy load level, and import block → quay crane area is the no-load level; in the A3 route, the quay crane area → export block level is the no-load level, and the export block → quay crane area is the heavy load level.

Model assumption

The assumptions of this paper are as follows:

  1. (1)

    There is only one container vessel in each dock, which can be loaded and unloaded by multiple quay cranes at the same time;

  2. (2)

    Each container truck can only transport one container;

  3. (3)

    Only the 20-foot scale container is considered;

  4. (4)

    The influence of quay crane position movement on truck transportation distance is not considered.

Level I: container truck routing optimization model without transportation job allocation

Model parameters

N:

The number of quay crane;

J:

The number of import block;

K:

The number of export block;

\(M_i\):

The number of container trucks allocated for the i th quay crane;

\(D_i\):

The number of containers to be unloaded by the i th quay crane;

\(H_i\):

The number of containers to be loaded by the i th quay crane;

\(\:d_{ij}^1\):

The distance from the quay crane i to the import block j;

\(\:d_{jk}^2\):

The distance from the import block j to the export block k;

\(\:d_{ki}^3\):

The distance from the export block k to the quay crane i;

\(\widetilde W_j^1\):

The container capacity of import block j, \(\widetilde W_j^1=U\left(\underline{w_j^1},\overline{w_j^1}\right)\);

\(\widetilde W_k^2\):

The container capacity of export block k; \(\widetilde W_k^2=U\left(\underline{w_k^2},\overline{w_k^2}\right)\);

\(\:{Z}_{ij}^{1}\):

Maximum number of container truck in the route of quay crane i and import block j;

\(\:{Z}_{ki}^{2}\):

Maximum number of container truck in the route of export block k and quay crane i;

\(\:{Z}_{jk}^{3}\):

Maximum number of container truck in the route of import block j and export block k;

\(\:{a}_{1}\):

The emissions per kilometer of heavy load container truck;

\(\:{a}_{2}\):

The emissions per kilometer of no-load container truck;

\(\:{b}_{1}\):

Diesel oil cost per kilometer of heavy load container truck;

\(\:{b}_{2}\):

Diesel oil cost per kilometer of no-load container truck;

\(T_1^0\):

Carbon quota for Level I;

\(\delta\):

Carbon trading price per ton;

\(\tau\):

Carbon tax per ton.

Decision variable:

\(X_{ijki'}\):

The number of container truck in the route of quay crane i → import block j → export block k → quay crane i’.

\(Y_{ij}^1\):

The number of container truck in the route of quay crane i→ import block j→ quay crane i.

\(Y_{ki}^2\):

The number of container truck in the route of export block k→ quay crane i→ export block k.

We realize the optimization of container truck’s path following two levels: In the first level, we try to optimize the transportation mission among quay crane area, import block, and export block based on the total transportation mission of container trucks. In the second level, having known the total mission from the first, we then allocate the missions to the trucks in an optimized way in order to ensure the most efficient transition of path between two missions, and to decide the number of trucks.

Model establishment of Level I

The total transportation cost of the container truck is divided into fuel cost, emission cost, and the emission quota profit.

The fuel cost is expressed as:

$$\begin{array}{l}C_{fuel}=b_1\left[\sum\:_{i^{'\:}=1}^N\sum\:_{k=1}^K\sum\:_{j=1}^J\sum\:_{i=1}^NX_{ijki^{'\:}}\left(d_{ij}^1+d_{ki^{'\:}}^3\right)+\sum\:_{j=1}^J\sum\:_{i=1}^NY_{ij}^1d_{ij}^1+\sum\:_{k=1}^K\sum\:_{i=1}^NY_{ki}^2d_{ki}^3\right]+b_2\left[\sum\:_{i^{'\:}=1}^N\sum\:_{k=1}^K\sum\:_{j=1}^J\sum\:_{i=1}^NX_{ijki^{'\:}}d_{jk}^2+\sum\:_{j=1}^J\sum\:_{i=1}^NY_{ji}^1d_{ji}^1+\sum\:_{k=1}^K\sum\:_{i=1}^NY_{ik}^2d_{ik}^3\right]\end{array}$$
(1)

Where \(\:\sum\:_{{i}^{{\prime\:}}=1}^{N}\sum\:_{k=1}^{K}\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{X}_{ijk{i}^{{\prime\:}}}\left({d}_{ij}^{1}+{d}_{k{i}^{{\prime\:}}}^{3}\right)\) is the number of heavy load container truck in the route of A1. \(\:\sum\:_{{i}^{{\prime\:}}=1}^{N}\sum\:_{k=1}^{K}\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{X}_{ijk{i}^{{\prime\:}}}{d}_{jk}^{2}\) is the number of no-load container truck in the route of A1. \(\:\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{Y}_{ij}^{1}{d}_{ij}^{1}\) is the number of heavy load container truck in the route of A2. \(\:\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{Y}_{ji}^{1}{d}_{ji}^{1}\) is the number of no-load container truck in the route of A2. \(\:\sum\:_{k=1}^{K}\sum\:_{i=1}^{N}{Y}_{ki}^{2}{d}_{ki}^{3}\) is the number of heavy load container truck in the route of A3. \(\:\sum\:_{k=1}^{K}\sum\:_{i=1}^{N}{Y}_{ik}^{2}{d}_{ik}^{3}\) is the number of no-load container truck in the route of A3.

The emission cost is a carbon tax levied by the government, expressed as:

$$\:{C}_{emission\:}=\tau\:{\left({T}_{1}^{0}-{a}_{1}\left[\sum\:_{{i}^{{\prime\:}}=1}^{N}\sum\:_{k=1}^{K}\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{X}_{ijk{i}^{{\prime\:}}}\left({d}_{ij}^{1}+{d}_{k{i}^{{\prime\:}}}^{3}\right)+\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{Y}_{ij}^{1}{d}_{ij}^{1}+\sum\:_{k=1}^{K}\sum\:_{i=1}^{N}{Y}_{ki}^{2}{d}_{ki}^{3}\right]-{a}_{2}\left[\sum\:_{{i}^{{\prime\:}}=1}^{N}\sum\:_{k=1}^{K}\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{X}_{ijki{\prime\:}}{d}_{jk}^{2}+\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{Y}_{ji}^{1}{d}_{ji}^{1}+\sum\:_{k=1}^{K}\sum\:_{i=1}^{N}{Y}_{ik}^{2}{d}_{ik}^{3}\right]\right)}^{+}$$
(2)

Where \(\:{\left(x\right)}^{+}=max\left(0,x\right)\). The emission quota profit is expressed as:

$$\:{R}_{trade}=\delta\:{\left({a}_{1}\left[\sum\:_{{i}^{{\prime\:}}=1}^{N}\sum\:_{k=1}^{K}\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{X}_{ijki{\prime\:}}\left({d}_{ij}^{1}+{d}_{ki{\prime\:}}^{3}\right)+\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{Y}_{ij}^{1}{d}_{ij}^{1}+\sum\:_{k=1}^{K}\sum\:_{i=1}^{N}{Y}_{ki}^{2}{d}_{ki}^{3}\right]+{a}_{2}\left[\sum\:_{{i}^{{\prime\:}}=1}^{N}\sum\:_{k=1}^{K}\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{X}_{ijki{\prime\:}}{d}_{jk}^{2}+\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{Y}_{ji}^{1}{d}_{ji}^{1}+\sum\:_{k=1}^{K}\sum\:_{i=1}^{N}{Y}_{ik}^{2}{d}_{ik}^{3}\right]-{T}_{1}^{0}\right)}^{+}$$
(3)

In this level, with the goal of minimizing the total driving cost of the truck, the t container truck routing optimization model without job allocation is established as follows:

$$\:{min}{TC}_{1}={C}_{fuel}+{C}_{emission\:}-{R}_{trade}$$
(4)

s. t.

$$\:{C}_{fuel}={\text{b}}_{1}\left[\sum\:_{{i}^{{\prime\:}}=1}^{N}\sum\:_{k=1}^{K}\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{X}_{ijk{i}^{{\prime\:}}}\left({d}_{ij}^{1}+{d}_{k{i}^{{\prime\:}}}^{3}\right)+\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{\text{Y}}_{\text{i}\text{j}}^{1}{d}_{ij}^{1}+\sum\:_{k=1}^{K}\sum\:_{i=1}^{N}{\text{Y}}_{\text{k}\text{i}}^{2}{d}_{ki}^{3}\right]+{\text{b}}_{2}\left[\sum\:_{{i}^{{\prime\:}}=1}^{N}\sum\:_{k=1}^{K}\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{X}_{ijk{i}^{{\prime\:}}}{d}_{jk}^{2}+\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{\text{Y}}_{\text{j}\text{i}}^{1}{d}_{ji}^{1}+\sum\:_{k=1}^{K}\sum\:_{i=1}^{N}{\text{Y}}_{\text{i}\text{k}}^{2}{d}_{ik}^{3}\right]$$
(5)
$$\:{C}_{emission\:}=\tau\:{\left({T}_{1}^{0}-{a}_{1}\left[\sum\:_{{i}^{{\prime\:}}=1}^{N}\sum\:_{k=1}^{K}\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{X}_{ijki{\prime\:}}\left({d}_{ij}^{1}+{d}_{ki{\prime\:}}^{3}\right)+\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{\text{Y}}_{\text{i}\text{j}}^{1}{d}_{ij}^{1}+\sum\:_{k=1}^{K}\sum\:_{i=1}^{N}{\text{Y}}_{\text{k}\text{i}}^{2}{d}_{ki}^{3}\right]+{a}_{2}\left[\sum\:_{{i}^{{\prime\:}}=1}^{N}\sum\:_{k=1}^{K}\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{X}_{ijki{\prime\:}}{d}_{jk}^{2}+\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{\text{Y}}_{\text{j}\text{i}}^{1}{d}_{ji}^{1}+\sum\:_{k=1}^{K}\sum\:_{i=1}^{N}{\text{Y}}_{\text{i}\text{k}}^{2}{d}_{ik}^{3}\right]\right)}^{+}$$
(6)
$$\:{R}_{trade}=\delta\:{\left({a}_{1}\left[\sum\:_{{i}^{{\prime\:}}=1}^{N}\sum\:_{k=1}^{K}\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{X}_{ijki{\prime\:}}\left({d}_{ij}^{1}+{d}_{ki{\prime\:}}^{3}\right)+\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{Y}_{ij}^{1}{d}_{ij}^{1}+\sum\:_{k=1}^{K}\sum\:_{i=1}^{N}{Y}_{ki}^{2}{d}_{ki}^{3}\right]+{a}_{2}\left[\sum\:_{{i}^{{\prime\:}}=1}^{N}\sum\:_{k=1}^{K}\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{X}_{ijki{\prime\:}}{d}_{jk}^{2}+\sum\:_{j=1}^{J}\sum\:_{i=1}^{N}{Y}_{ji}^{1}{d}_{ji}^{1}+\sum\:_{k=1}^{K}\sum\:_{i=1}^{N}{Y}_{ik}^{2}{d}_{ik}^{3}\right]-{T}_{1}^{0}\right)}^{+}$$
(7)
$$\:\sum\:_{i^{'\:}=1}^N\sum\:_{k=1}^K\sum\:_{j=1}^JX_{ijki'\:}+\sum\:_{j=1}^JY_{\mathrm{ij}}^1\geq\:D_i,\:i\epsilon N$$
(8)
$$\:\sum\:_{k=1}^K\sum\:_{j=1}^J\sum\:_{i=1}^NX_{ijki'\:}+\sum\:_{k=1}^KY_{\mathrm{ki}'\:}^2\geq\:H_{i'\:},\:i'\:\epsilon N$$
(9)
$$\:p\left(\sum\:_{i^{'\:}=1}^N\sum\:_{k=1}^K\sum\:_{i=1}^NX_{ijki'\:}+\sum\:_{i=1}^NY_{ij}^1\leq\:\widetilde W_j^1\right)\geq\:f_j^1,\:j\epsilon J$$
(10)
$$\:p\left(\sum\:_{i^{'\:}=1}^N\sum\:_{j=1}^J\sum\:_{i=1}^NX_{ijki'\:}+\sum\:_{i=1}^NY_{\mathrm{ki}}^2\leq\:\widetilde W_k^2\right)\geq\:f_k^2,\:k\epsilon K$$
(11)
$$\:\sum\:_{k=1}^K\sum\:_{i'\:=1}^NX_{ijk\mathrm i'\:}+Y_{ij}^1\leq\:Z_{ij}^1,\:i\epsilon N,\:j\epsilon J$$
(12)
$$\:\sum\:_{i=1}^N\sum\:_{j=1}^JX_{ijk\mathrm i'\:}+Y_{ij}^1\leq\:Z_{ki'\:}^2,\:i'\epsilon N,\:k\epsilon K$$
(13)
$$\:\sum\:_{i=1}^N\sum\:_{i'\:=1}^NX_{ijk\mathrm i'\:}\leq\:Z_{jk'\:}^3,\:j\epsilon J,\:k\epsilon K$$
(14)
$$\:X_{ijk\mathrm i'\:}\geq\:0,\:i,i'\:\epsilon N,\:j\epsilon J,\:k\epsilon K$$
(15)
$$\:Y_{ij}^1\geq\:0,\:i\epsilon N,\:\:j\epsilon J$$
(16)
$$\:Y_{ki}^2\geq\:0,\:i\epsilon N,\:k\epsilon K$$
(17)

The objective function (4) represents the minimum total cost of container truck transportation; Eq. (5) represents the fuel cost; Eq. (6) represents the emission cost; Eq. (7) represents the emission quota profit; Eq. (8) represents that the unloading requirements of each quay crane are met; Eq. (9) represents that the loading requirements of each quay crane are met; Eq. (10) represents the probability of meeting the import block capacity limit, where p (x) represents the probability of x occurring; Eq. (11) represents the probability of meeting the export block capacity limit; Eq. s (12) to (14) represent the upper limit of the number of container trucks between quay crane, import block and export block, which is the congestion limit; Eq. s (15) to (17) represent the range of decision variable values.

Level II: the container trucks no-load routing optimization model of transportation job allocation

According to Level I, it can be seen that the container trucks have three route choices of A1, A2 and A3. After each container truck completes a transportation route job, the next transportation job can be selected. Since the transition between tasks can lead to the status of no-load of trucks, the order of tasks being fulfilled also needs to be optimized. We choose a transport route for each truck as a task. We set each container truck to choose a transportation route as a job. This level is the optimization of the container truck transportation job, so the container trucks are all no-load.

Model parameters

S:

The set of container truck transportation job, according to Level I, the number of job can be calculated as Q, r,\(\:u\)=1, 2, …, Q;

H:

The set of container trucks. The number of container trucks is h, and m represents the container truck serial number, m = 1, 2, …, h;

\(L_{ru}\):

The distance between job r and job u;

\(T_r\):

The transportation time of job r;

\(T_{ru}^m\):

The time for container truck m to execute job r and then job u;

\(L_i\):

The loading and unloading time limit of quay crane i;

\(T_2^0\):

Carbon quota for Level II.

Decision variable:

\(F_{ru}^m\):

If container truck m executes job r before executing job u, then \(F_{ru}^m=1\), otherwise, \(F_{ru}^m=0\).

 \(F_r^m\):

If container truck m performs job r, then \(F_r^m=1\), otherwise \(F_r^m=0\).

Model establishment of Level II

The container trucks’ no-load routing optimization model of transportation job allocation is as follows:

$$\:{min}{TC}_{2}={b}_{2}\sum\:_{r\in\:S}\sum\:_{u\in\:S}\sum\:_{m\in\:H}{L}_{ru}{F}_{ru}^{m}+\tau\:{\left({a}_{2}\sum\:_{r\in\:S}\sum\:_{u\in\:S}\sum\:_{m\in\:H}{L}_{ru}{F}_{ru}^{m}-{T}_{2}\right)}^{+}-\delta\:{\left({T}_{2}^{0}-{a}_{2}\sum\:_{r\in\:S}\sum\:_{u\in\:S}\sum\:_{m\in\:H}{L}_{ru}{F}_{ru}^{m}\right)}^{+}$$
(18)

s.t.

$$\:\:{\textstyle\sum_{m\in\:H}}\;F_r^m=1,\:r\epsilon S$$
(19)
$$\:{\textstyle\sum_{r\in\:S}}\:F_{ru}^m=F_u^m,\:m\epsilon H,u\epsilon S$$
(20)
$$\:{\textstyle\sum_{u\in\:S}}\:F_{ru}^m=F_r^m,\:m\epsilon H,r\epsilon S$$
(21)
$$\:{\textstyle\sum_{u\in\:S}}\:T_rF_r^m+{\textstyle\sum_{r\in\:S}}\:{\textstyle\sum_{u\in\:S}}\:T_{ru}^mF_{ru}^m\leq\:minL_i,\:\forall\:\:m\in\:H,i\in\:N$$
(22)
$$\:F_{ru}^m\epsilon\left\{0,1\right\},\:m\epsilon H,\:r,u\epsilon S$$
(23)
$$\:F_r^m\epsilon\left\{0,1\right\},\:\:m\epsilon H,\:r\epsilon S$$
(24)
$$\:F_u^m\epsilon\left\{0,1\right\},\:\:m\epsilon H,\:u\epsilon S$$
(25)

The objective function (18) represents the minimum total cost of container truck transportation; \(\:{b}_{2}\sum\:_{r\in\:S}\sum\:_{u\in\:S}\sum\:_{m\in\:H}{L}_{ru}{F}_{ru}^{m}\) is fuel cost; \(\:\tau\:({a}_{2}\sum\:_{r\in\:S}\sum\:_{u\in\:S}\sum\:_{m\in\:H}{L}_{ru}{F}_{ru}^{m}\)\(\:-{T}_{2}^{0}{)}^{+}\) is the emission cost;\(\:\:\delta\:{\left({T}_{2}-{a}_{2}\sum\:_{r\in\:S}\sum\:_{u\in\:S}\sum\:_{m\in\:H}{L}_{ru}{F}_{ru}^{m}\right)}^{+}\) is the emission quota profit; Eq. (19) represents that each job is completed by a container truck; Eq. s (20) and (21) are the relationship between decision variables; Eq. s (22) represents the time limit for container loading and unloading of quay crane i; Eq. s (23) to (25) are the decision variable value constraints.

Model solving

The solution to the container truck routing optimization problem has considerable time and space complexity. Therefore, the complexity of the container truck routing optimization problem will be further improved, when the relationship between different links is considered. How to realize the plan coordination between each link of the operation system, and effectively deal with the relationship between various constraints to improve the efficiency of model solving, has always been a difficult problem of container terminal operation plan coordination. For these two models, we plan to use the improved PSO to solve them. The traditional PSO is easy to fall into the trap of local optimal solution. In this paper, the initial solution space of PSO is optimized to improve the accuracy of the algorithm, and the PSO is improved according to the characteristics of the container truck routing optimization model. The specific steps are as follows:

Step 1:

initialize N particles, x10, x20,…, xN0, then the model solution space is decomposed into N regions.

Step 2:

M particles are randomly generated in \(\left[x_{i0}-\delta,\;x_{i0}+\delta\right]\) of each particle \(x_{i0}\). In order not to disturb other particles, let \(\delta=\frac{l_u-l_d}{2N}\), where \(l_u\)  and \(l_d\) are the upper and lower limits of \(x_{i0}\), respectively. \(x_{ij}\) represents the j th particle generated by the particle  as the center. Since there are three routes in the Level I, route A1 needs N·J·K·N particles, route A2 needs N·J particles, and route A3 needs N·K particles, so M = N·J·K·N + N·J + N·K. In order to ensure that the particles are integers greater than or equal to 0, we let, and the function \(round\left(x_{ij}\right)\)  is the integer closest to \(x_{ij}\).

Step 3:

the particle velocity in each region is calculated as follows.

$$\:{v}_{ij}=w\cdot\:{v}_{ij}+c1\cdot\:rand\cdot\:\left({p}_{ij}-{x}_{ij}\right)+c2\cdot\:rand\cdot\:\left({g}_{i}-{x}_{ij}\right)$$
(26)
$$\:{x}_{ij}={x}_{ij}+{v}_{ij}$$
(27)

\(\:rand\) is a random number between (0,1), \(\:c1\) and \(\:c2\) are learning factors, \(\:w\) is the weighting coefficient, \(\:{p}_{ij}\) is the best position in the search process of the j particles in the i th region so far, and \(\:{g}_{i}\) is the best position in the i th region as a whole. Step 3 is iterated to calculate \(\:{Z}_{1}\) times.

Step 4:

return step 2, let \(x_{i0}=g_i\), and calculate Z2 times from step 2 to step 4.

Step 5:

the best particles \(x_i\) in each region are selected respectively, and the classical particle swarm optimization algorithm is applied to these N particles to obtain the optimal solution.

$$\:{v}_{i}=w\cdot\:{v}_{i}+c1\cdot\:rand\cdot\:\left({p}_{i}-{x}_{i}\right)+c2\cdot\:rand\cdot\:\left({g}_{best}-{x}_{i}\right)$$
(28)
$$\:{x}_{i}={x}_{i}+{v}_{i}$$
(29)

\(\:{p}_{i}\) is the best position in the search process of the i th particle so far, \(\:{g}_{best}\) is the best position in the whole, and \(\:{Z}_{3}\) times are calculated iteratively. Our process of improved PSO algorithm is shown in Fig. 2.

Figure 2
figure 2

Process of improved PSO algorithm.

Full size image

For the calculation of Eq. (10) and Eq. (11), since \(\:p\left(x\le\:{\text{W}}_{\text{j}}^{1}\right)\)=\(\:1-{F}_{{\text{W}}_{\text{j}}^{1}}\left(x\right)\), \(\:{F}_{{\text{W}}_{\text{j}}^{1}}\:\)is the probability distribution function of uniformly distributed \(\:U(\underset{\_}{{w}_{j}^{1}},\overline{{w}_{j}^{1}})\), the specific expression is shown in Eq. (30).

$$\:F_{W_{\mathrm j}^1}\left(x\right)=\left\{\begin{array}{cc}0&\:x\leq\:\underset\_{w_j^1}\\\:\frac{x-\underset\_{w_j^1}}{\overline{w_j^1}-\underset\_{w_j^1}}&\:1<x<\overline{w_j^1}\\\:1&\:x\geq\:\overline{w_j^1}\end{array}\right.$$
(30)

According to Eq. (30), the probability constraints of Eq. (10) and Eq. (11) can be transformed into piecewise deterministic constraints.

Example analysis

Example data

In order to verify the effectiveness of the models and algorithm, this section uses data from one container terminal for example analysis. The port which we choose to analyze is Zhifu Bay Port in Shandong Province, China. This port is the origin and the present core area of Yantai Port, which is located in the northern part of central area of Yantai City. There are a total of 59 berths of various types, 36 deep-water berths with a capacity of over 10,000 tons, and a total length of 11,268 m for the dock shoreline.

There are two quay cranes in the container terminal yard, two export blocks and three import blocks. The distances between the import block, the export block and the quay crane, as well as the upper limit of the number of container trucks are shown in Table 1.

Table 1 The distances between quay cranes, import and export block, and maximum number of container trucks (distances (km), maximum number of container trucks).
Full size table

The container loading and unloading capacities of quay crane 1 are 85 TEU (Twenty-foot Equivalent Unit) and 60 TEU respectively, and the container loading and unloading capacities of quay crane 2 are 54 TEU and 81 TEU respectively. The capacities of export block and import block are evenly distributed, and the specific distribution function is shown in Table 2.

Table 2 The capacity of export block and import block (TEU).
Full size table

By investigating Zhifu Bay Port, we learned that the diesel oil consumption of the heavy load container truck is 1.2 L/km, the no-load diesel oil consumption is 0.8 L/km, the fuel emission coefficient is 2.65 kg/L, and the diesel price is 1$/t. The carbon trading price and the carbon tax are different in different regions. As for the carbon trading price, we use Statista ‘s European Union price on March 31,2023, which is 96$/t40. For the carbon tax we adopt IMO’s suggested price 100$/t41. The carbon emission of the heavy load container truck is 3.18E-03t/km, and the no-load carbon emission is 2.12E-03t/km. The probability of meeting the import block capacity limit is set to \(\:{f}_{1}^{1}={f}_{2}^{1}={f}_{2}^{1}=\) 0.8, and the probability of meeting the export block capacity limit is set to \(\:{f}_{1}^{2}={f}_{2}^{2}=\) 0.8, \(\:{T}_{1}^{0}\)= 0.50t. As for the tool and environment of this experiment, all of our test functions and algorithms were coded in a MATLAB R2016a (64-bit) numerical calculation software, and implemented on an Intel (R) Core (TM) i5-7400 CPU @ 3.00 GHz 3.00 GHz with 8 GB of RAM, and the experimental environment is the version of Windows 10.

Analysis and calculation of Level I

In this section we use the improved PSO algorithm to calculate container truck routing optimization model, and the results are shown in Table 3. \(\:{X}_{1211}\)=40 indicates that the task from the route of quay crane 1 → import block 2 → export block 1 → quay crane 2 is 40 times, and there are 40 containers of loading and 40 containers of unloading in quay crane (1) \(\:{X}_{2211}\)=26 indicates that the task of the route quay crane 2 → import block 2 → export block 1 → quay crane 1 is 26 times, and there are 25 containers of loading in quay crane 1, as well as 26 containers of unloading in quay crane (2) \(\:{X}_{2322}\)=33 indicates that the task of the route quay crane 2 → import block 3 → export block 2 → quay crane 2 is 33 times, and there are 33 containers of loading and 33 containers of unloading in quay crane 2. \(\:{Y}_{22}^{1}=\)22 indicates that the task of the route quay crane 2 → import block 2 → quay crane 2 is 22 times, and there are 22 containers of loading and 22 containers of unloading in quay crane 2. \(\:{Y}_{11}^{2}\)= 20 indicates that the task of route quay crane 1 → import block 1 → quay crane 1 is 20 times, and there are 20 containers of loading and 20 containers of unloading in quay crane 1. See the results of Level 1 in Table 3.

Table 3 Calculation results of Level I.
Full size table

The total cost of Level I \(\:{TC}_{1}\)=422.21$, while \(\:{C}_{fuel}\)=373.29$, \(\:{C}_{emission\:}\)=48.92$, and =0. The total transportation distance is 345.01 km, including the heavy load of 243.21 km and an empty load of 101.80 km, with carbon emission being 0.989t.

Analysis and calculation of Level II

We set the number of trucks to be h = 11, the carbon emission quota for Level II \(\:{T}_{2}^{0}\)=1.00E-02t, and the loading and unloading time limit for quay crane 1 and quay crane 2 are both L2=\(\:{L}_{2}=\)130 minutes. The transportation speed of the container truck is 20 km/h, and the task conversion time \(T_{ru}^m\)=1 min. The distance between each transportation task of each truck is shown in Table 4.

Table 4 Transportation distance and conversion distance for each task (km).
Full size table

We still used the improved PSO algorithm to solve the model. We can calculate the task sequence for each container truck, as shown in Table 5. For example, the total number of tasks accepted by the container truck 1 is 13, in the order of 1 → 1 → 5 → 5 → 2 → 5 → 2 → 1 → 3 → 1 → 5 → 1 → 3. The calculation results show that the total cost of task conversion in Level II \(\:{TC}_{2}\)=10.13$, with fuel cost of 8.8 $, emission cost of 1.33 $, and emission quota profit of 0$. The total distance of the container truck task conversion is 11 km, the fuel consumption is 8.8 L, and the carbon emissions is 2.33E-02t. The task sequence and total time consumption of container trucks are shown in Table 5.

Table 5 Task sequence and total time consumption of container trucks.
Full size table

Sensitivity analysis of carbon tax

The impact of carbon tax changes on emissions in the two levels are shown in Fig. 3:

Figure 3
figure 3

The impact of carbon tax changes on emissions in two levels.

Full size image

According to Fig. 3, we can see that the carbon emissions in both levels exceed their quotas, so the port has to pay carbon taxes to the government. With the increase of carbon tax, the emission in Level I is gradually decreasing, but there is almost no change in Level II. This is because in Level II, all the container trucks are empty, and the main purpose of optimizing the allocation of empty transportation tasks for container trucks is to minimize the transportation distance for container truck task conversion. The effect of increasing carbon tax is consistent with the effect of increasing fuel price, and it will not shorten the transportation distance for container truck task conversion, so it will not reduce the emissions.

Sensitivity analysis of number of container trucks

The number of container trucks has a significant impact on the choice of transportation strategy of containers in port as well as on their transportation cost. Therefore, it is quite necessary to analyze the number of container trucks. It can be seen that as the number of container trucks increases, the truck task conversion distance for Level II decreases, as is shown in Fig. 4. The more container trucks there are, the better the port can arrange container truck transportation work, but it may lead to an increase in transportation costs. The impact of the number of trucks on the transportation distance in Level II is shown in Fig. 4.

Figure 4
figure 4

The impact of the number of trucks on the transportation distance in Level II.

Full size image

Conclusion

Reducing the emissions of container trucks is an important means of building low-carbon ports. This paper analyzed the routes of container trucks between the Quay crane and the import & export blocks. Based on the characteristics of the routes of container trucks, three types of driving routes are analyzed. Considering the capacity of import and export blocks as uncertain variables, we constructed a container truck routing optimization model with the goal of minimizing total cost without considering transportation task conversion, as well as a container truck empty routing optimization model with the goal of minimizing task allocation cost. The government also implements carbon emission quota policies for container truck transportation.

We designed an improved PSO algorithm to solve these two models, calculated the task order of each container truck, and analyzed the impact of carbon tax on emissions in two levels respectively, as well as the impact of changes in the number of container trucks on the transportation distance of Level II. The result showed that an increase in carbon tax will lead to a decrease in emissions in Level I, but has no impact on emissions in Level II; The increase in the number of container trucks will lead to a reduction in the transportation distance for Level II. Compared to the previous researches, our method was brought up especially to solve the problem of port container trucks’ route optimization, thus more pertinent and targeted, and it is proved to have better performance and optimization effects.

This study only considered the situation where each container truck can only transport one container, but in actual business, there are also container trucks that can transport two containers at the same time. Therefore, in future research, the optimization problem of scheduling where a single transportation task of a container truck can transport multiple containers can be considered.