Optimization of Multimodal Paths for Oversize and Heavyweight Cargo under Different Carbon Pricing Policies

Abstract

With the increasing global concern over climate change, reducing greenhouse gas emissions has become a universal goal for governments and enterprises. For oversize and heavyweight cargo (OHC) transportation, multimodal transportation has become widely adopted. However, this mode inevitably generates carbon emissions, making research into effective emission reduction strategies essential for achieving low-carbon economic development. This study investigates the optimization of multimodal transportation paths for OHC (OMTP-OHC), considering various direct carbon pricing policies and develops models for these paths under the ordinary scenario—defined as scenarios without any carbon pricing policies—and two carbon pricing policy scenarios, namely the emission trading scheme (ETS) policy and the carbon tax policy, to identify the most cost-effective solutions. An enhanced genetic algorithm incorporating elite strategy and catastrophe theory is employed to solve the models under the three scenarios. Subsequently, we examine the impact of ETS policy price fluctuations, carbon quota factors, and different carbon tax levels on decision-making through a case study, confirming the feasibility of the proposed model and algorithm. The findings indicate that the proposed algorithm effectively addresses this problem. Moreover, the algorithm demonstrates a small impact of ETS policy price fluctuations on outcomes and a slightly low sensitivity to carbon quota factors. This may be attributed to the relatively low ETS policy prices and the characteristics of OHC, where transportation and modification costs are significantly higher than carbon emission costs. Additionally, a comparative analysis of the two carbon pricing policies demonstrates the varying intensities of emission reductions in multimodal transportation, with the ranking of carbon emission reduction intensity as follows: upper-intermediate level of carbon tax > intermediate level of carbon tax > lower-intermediate level of carbon tax = ETS policy > the ordinary scenario. The emission reduction at the lower-intermediate carbon tax level (USD 8.40/t) matches that of the ETS policy at 30%, with a 49.59% greater reduction at the intermediate level (USD 50.48/t) compared to the ordinary scenario, and a 70.07% reduction at the upper-intermediate level (USD 91.14/t). The model and algorithm proposed in this study can provide scientific and technical support to realize the low-carbonization of the multimodal transportation for OHC. The findings of this study also provide scientific evidence for understanding the situation of multimodal transportation for OHC under China’s ETS policy and its performance under different carbon tax levels in China and other regions. This also contributes to achieving the goal of low-carbon economic development.

1. Introduction

Oversize and heavyweight cargoes (OHC), comprising mainly large-scale devices for industrial, agricultural, and military purposes, play a crucial role in supporting key national projects, national defense construction, and promoting the development of the national economy [1]. Currently, as the application of large-scale power equipment, chemical machinery, energy infrastructure, and marine engineering equipment becomes more widespread, the need for OHC transportation has significantly increased. Given the challenges of meeting OHC transportation needs through a single mode, multimodal transportation has become the preferred method for moving OHC [2]. Multimodal transport, which combines various modes of transport, aims to reduce costs and shorten transit times, thereby enhancing efficiency. This method has seen widespread adoption and development globally, yet it confronts numerous challenges and issues. A significant concern involves managing the impacts of diverse carbon emission policies and striving for the low-carbonization of multimodal transport.

With the increasing impact of climate change, countries worldwide have adopted two primary policies for direct carbon pricing: a carbon tax and an Emissions Trading Scheme (ETS). These policies employ diverse pricing mechanisms to promote emission reductions and reduce greenhouse gas emissions. The carbon tax policy involves a fixed tax levied by the government on greenhouse gas emissions released into the atmosphere by enterprises, termed as the carbon tax rate (USD/t of CO₂). The carbon tax rate can be adjusted in response to government policy. The ETS policy analyzed in this paper represents a cap-and-trade system; enterprises are allocated a specific carbon emission quota. Should their emissions exceed this quota, they may purchase additional permits at the prevailing market rate (USD/t of CO₂), or sell surplus permits if applicable.

Regarding carbon tax policies, some regions and countries, such as Canada and Sweden, have already included the transportation sector within their carbon taxation frameworks, while most other nations apply carbon taxes primarily to industries like manufacturing and power generation. And China has not yet implemented a comprehensive carbon tax system.

In terms of ETS policies, there are currently 36 carbon emission trading systems operational worldwide, covering major industries including power, industry, and aviation. Starting in 2024, the EU’s ETS plans to extend coverage to the maritime industry. In China, the ETS was initially implemented in the power sector, with an expectation to cover 72% of the nation’s carbon emissions by 2025 [3]. Heavy trucks have maintained a steady growth in ownership over the past few years, with approximately 8 million units owned by 2020, growing annually by 7%. These vehicles account for 11.4% and 15.9% of carbon monoxide and hydrocarbon emissions, respectively, among all vehicle types [4]. Therefore, heavy trucks are major contributors to pollutant emissions in the transportation sector, and addressing truck emissions is crucial for achieving carbon peak and neutrality goals.

Given that OHC typically requires transportation via heavy trucks, this directly re-lates to the carbon emission issues of OHC transportation. Thus, researching the OMTP-OHC, optimizing its multimodal transportation paths, and understanding the performance of the multimodal transportation of OHC under various carbon pricing policies are both reasonable and necessary. This may also contribute to strategies aimed at achieving carbon peaking and neutrality.

However, research on the impact of these policies on multimodal transportation for OHC remains insufficient. Specifically, there is a lack of relevant studies evaluating the specific effects of carbon tax price and current ETS prices on decision-making.

Based on this, this article aims to explore the impact of carbon tax policy and current ETS policy on the OMTP-OHC. It analyzes the effects of different price levels of carbon taxes and ETS prices under current market fluctuations on the costs and carbon emissions of OHC transportation. It can provide a reference for the low-carbonization of the multimodal transportation of OHC.

2. Literature Review

OHCs are frequently transported using multimodal transportation methods [2]. Early research primarily focused on identifying and discussing the influential factors affecting OHC transportation. For instance, Petraška and Jarasuniene [5] and Lei [6] analyzed factors such as turning radius, transshipment capacity, and bridge bearing capacity in OHC transportation processes. Currently, there is significant attention directed toward multimodal OHC transportation issues related to the reconstruction of lines and nodes. Researchers like Luo et al. [1] and Lei [6] contend that the reconstruction of lines and nodes is a crucial component of multimodal OHC transportation. Additionally, due to the unique characteristics of OHC, Zhang et al. [7,8] quantified the risks associated with OHC transportation and assessed the impact of road segments on regular cargo transportation. In sum, academic research on route decision-making for the multimodal transportation of OHC often revolves around minimizing transport costs, time, and risk, with fewer researchers focusing on carbon pricing policies for the multimodal transportation of OHC.

The path-optimization problem of multimodal transportation is often solved using metaheuristic algorithms and shortest-path algorithms [9,10]. The common metaheuristic algorithms are genetic algorithms [11,12] and ant colony algorithms [13,14]. Each of these algorithms has its own strengths and weaknesses. Metaheuristic algorithms can ensure high efficiency and fast solution speed but may occasionally become trapped in local optimal solutions, making it difficult to obtain global optimal solutions. On the other hand, the advantages of shortest-path algorithms lie in their high solution accuracy, ability to ensure optimality, and good solution stability. However, they are time-consuming and inefficient for solving complex problems, especially for ultra-large-scale problems. By enhancing heuristic algorithms using operators to escape from local optimal solutions, it is possible to maintain solution stability and superiority while ensuring high efficiency. Therefore, this study designs a catastrophe operator and elite retention strategy to improve genetic algorithms.

With the recognition of the greenhouse effect, the concepts of low-carbon environmental protection and energy conservation have gradually gained widespread acceptance in society. Large-scale greenhouse gas emissions have attracted widespread attention in the logistics industry. Consequently, an increasing number of scholars have begun to explore the impact of carbon emissions on multimodal transportation route decision-making. The development of research on multimodal transportation route optimization from a low-carbon perspective has progressed rapidly. Liao et al. [15] explored and demonstrated the environmental advantages of multimodal transportation modes and routes over single transportation modes. Many scholars have approached the problem from the perspective of carbon reduction, considering transportation carbon emissions or emission costs as objectives or constraints, and establishing multi-objective optimization models based on total cost, total time, total carbon emissions, etc. [16,17,18,19]. Furthermore, scholars have found that the carbon emission volume can be internalized as part of the cost. Therefore, they have used the total cost including carbon emission costs as the objective function for model optimization [20,21,22,23]. For example, Zhang et al. [24] points out that under a competitive supply chain model, the manufacturer’s emission reduction investment strategy strikes a balance between the economy and the environment, and that the retailer’s capital constraints can facilitate the manufacturer’s emission reduction investment. This echoes cost and carbon considerations in multimodal transportation path optimization. Xiao et al. [25] examines one of the key tools for achieving carbon neutrality, BEBs, and analyzes the economics of electric buses with varying charging rates, and proposes a total cost of ownership (TCO) framework. Zhang et al. [26] found that the total system cost can be reduced through economies of scale when the emission price of CO₂ is increased. Moreover, when the price of carbon dioxide emissions increases, system total costs can be reduced through economies of scale. Additionally, some researchers have analyzed multimodal transportation route optimization problems under the decision-making con-text of carbon pricing policies, studying and designing various calculation formulas and models for carbon emissions, investigating the carbon reduction efforts of different carbon pricing policies, and their impact on route decisions [19,27,28,29,30,31,32,33,34]. In low-carbon research on multimodal transportation, capacity and time uncertainties are often considered. Researchers model uncertainty factors, study path decisions and their impact on carbon emissions under uncertain conditions, and typically employ methods such as fuzzy numbers and random numbers to solve uncertainties and address problems [35,36,37,38]. In summary, scholars have conducted numerous studies in the field of low-carbon multimodal transportation route optimization, but few have investigated the impact of current carbon pricing policies on decision-making for OHC.

In summary, heuristic algorithms for solving these types of problems are well developed and have yielded significant results. However, research on the OMTP-OHC under various policies is relatively scarce. Specifically, there are few studies on the impact of fluctuations in China’s actual ETS policy, as well as the effects of different levels of carbon tax on the OMTP-OHC. Given that these issues accurately reflect the outcomes of path optimization under real carbon pricing policy implementation scenarios, they possess strong practicality and hold significant application value in the field of the OMTP-OHC.

3. Materials and Methods

The multimodal transportation network for OHC comprises three modes: highway, railway, and waterway. Carriers are required to deliver the goods from origin to destination within the customer-specified maximum delivery time. Intermediate nodes between the origin and destination facilitate multimodal transfers. The OHCs are characterized by their high value, significant weight, and large volume. Modifications, such as route clearances at nodes and increased transshipment capacity, may be necessary in certain parts of the transportation network due to the significant weight and volume of the goods, which contribute to higher carbon emissions. Furthermore, the larger machinery and vehicles required also contribute to increased carbon emissions. Consequently, compared to general cargo transportation, multimodal OHC transportation incurs higher carbon emissions and increased costs.

Since different carbon pricing strategies maybe significantly affect costs, this research includes various carbon pricing policies within the OMTP-OHC framework, focusing on minimizing costs under these diverse policy scenarios.

The study is grounded on the following assumptions:

• The OHC is indivisible during transportation and must be handled as complete units without the option of being subdivided.
• Waiting time and associated costs incurred during cargo transshipment are excluded from consideration.
• Factors such as weather fluctuations, traffic congestion, cargo damage, and equipment malfunction during transportation are not factored into the analysis.
• Transshipment exclusively takes place at designated nodes, with a maximum allowance of one transshipment per node.

The notations introduced in

Table 1. Description of notations.

Set	Description
$V$	The set of nodes in the transportation network $V=\{v_1,v_2,v_3,\dots ,v_n\}$
$W$	The transportation mode set, $\left\{W_a\right|\mathrm{1,2},3\}$, where $a=\mathrm{1,2},3$, represents, respectively, the modes of road, rail, and water transportation
$R$	The set of transportation links between adjacent nodes, denoted as $R$$=\left\{r_{ij}\right|v_i,v_j\in V$}
$O=\{\mathrm{V},\mathrm{W},\mathrm{R}\}$	The network for the transportation of OHC
$B\left(i\right)$	The set of predecessor nodes of $v_i$$, B\left(i\right)ϵR$
$A\left(i\right)$	$\mathrm{The} \mathrm{set} \mathrm{of} \mathrm{successor} \mathrm{nodes} \mathrm{of} v_i$$, A\left(i\right)ϵR$
$c_{ija}^1$	The transportation cost incurred when transporting from node $i$$\mathrm{to} \mathrm{node} j$ $\mathrm{using} \mathrm{transportation} \mathrm{mode} W_a/USD$
$c_{ija}^1$	The modification cost incurred when transporting from node $i$$\mathrm{to} \mathrm{node} j$ $\mathrm{using} \mathrm{transportation} \mathrm{mode} W_a/USD$
$C_e$	ETS price$/USD$
$C_{tax}$	$\mathrm{Carbon} \mathrm{tax} \mathrm{price}/USD$
$E_p$	$\mathrm{Carbon} \mathrm{emission} \mathrm{quota} /t$
$Q_a$	$\mathrm{The} \mathrm{total} \mathrm{weight} \mathrm{of} \mathrm{the} \mathrm{OHC} \mathrm{and} \mathrm{vehicle} \mathrm{when} \mathrm{transporting} \mathrm{OHC} \mathrm{via} \mathrm{transportation} \mathrm{mode} W_a/t$
$e_{ija}^1$	The carbon emissions generated when transporting from node $i$$\mathrm{to} \mathrm{node} j$$\mathrm{using} \mathrm{transportation} \mathrm{mode} W_a/t$
$e_{ija}^2$	The carbon emissions generated during the transportation from node $i$$\mathrm{to} \mathrm{node} j$$\mathrm{using} \mathrm{transportation} \mathrm{mode} W_a/t$
$d_{ija}$	The transportation distance from from node $i$$\mathrm{to} \mathrm{node} j$$\mathrm{using} \mathrm{transportation} \mathrm{mode} W_a/km$
$e_{iab}$	$\mathrm{The} \mathrm{carbon} \mathrm{emissions} \mathrm{generated} \mathrm{from} \mathrm{changing} \mathrm{the} \mathrm{transportation} \mathrm{mode} \mathrm{from} W_a$$\mathrm{to} W_b$$\mathrm{at} \mathrm{node} i/t$
$G_{ija}$	The carrying capacity of the bridge after modification when using transportation mode $W_a$ to traverse node $i$ to node $j$
$J_{ija}$	$\mathrm{The} \mathrm{clearance} \mathrm{of} \mathrm{the} \mathrm{route} \mathrm{after} \mathrm{modification} \mathrm{when} \mathrm{using} \mathrm{transportation} \mathrm{mode}{ W}_a$ to traverse node $i$ to node $j$
$J_a$	The combined profile of the vehicle and cargo after loading when transporting with transportation mode $W_a$
$H_{iab}$	$\mathrm{The} \mathrm{transformation} \mathrm{capacity} \mathrm{at} \mathrm{node} i$$\mathrm{when} \mathrm{switching} \mathrm{from} \mathrm{transportation} \mathrm{mode} W_a$$\mathrm{to} W_b$
$t_{ija}$	The transportation time from node $i$ to node $j$$\mathrm{using} \mathrm{transportation} \mathrm{mode} W_a/h$
$t_{iab}$	$\mathrm{The} \mathrm{transportation} \mathrm{time} \mathrm{for} \mathrm{switching} \mathrm{from} \mathrm{mode} W_a$$\mathrm{to} W_b$$\mathrm{at} \mathrm{node} i/h$
$T$	The maximum acceptable time for the carrier
$S$	$\mathrm{The} \mathrm{carbon} \mathrm{emission} \mathrm{quota} \mathrm{traded} \mathrm{by} \mathrm{market} \mathrm{participants}, \mathrm{where} S<0$$\mathrm{indicates} \mathrm{selling} \mathrm{one}’\mathrm{s} \mathrm{own} \mathrm{carbon} \mathrm{emission} \mathrm{quota} \mathrm{for} \mathrm{profit}, \mathrm{and} S>0$ indicates buying carbon emission quotas on the market, incurring costs
$x_{ija}$	$0-1$ variable, 1 if $\mathrm{uses} \mathrm{transportation} \mathrm{mode} W_a$$\mathrm{to} \mathrm{transport} \mathrm{from} \mathrm{node} i$$\mathrm{to} \mathrm{node} j$ and $0$, otherwise
$y_{iab}$	1 if $\mathrm{switches} \mathrm{from} \mathrm{transportation} \mathrm{mode} W_a$$\mathrm{to} \mathrm{mode} W_b$$\mathrm{at} \mathrm{node} i$ and $0$, otherwise

below were used to describe this problem:

3.1. Ordinary Modal

Based on the above comprehensive considerations, the OMTP-OHC model under the ordinary scenario ($M1$)—which refers to conditions without any carbon pricing policies—is constructed as follows with the optimization objective of minimizing the total vehicle transportation cost:

$$min{Z}_1=\sum _{a=1}^3\sum _{i=1}^n\sum _{j=1}^n{x}_{ija}\left(c_{ija}^1+c_{ija}^2\right)+\sum _{a=1}^3\sum _{b=1}^3\sum _{i=1}^n{y}_{iab}\left(c_{iab}^1+c_{iab}^2\right)$$

(1)

which is subject to the following:

$$x_{ija}{Q}_a\le G_{ija},v_i,v_j\in V,W_a\in W$$

(2)

$$x_{ija}{J}_a\le J_{ija},v_i,v_j\in V,W_a\in W$$

(3)

$$y_{iab}\le H_{iab},v_i\in V,W_a,W_b\in W$$

(4)

$$\sum _{a=1}^3{x}_{ija}\le 1,v_i,v_j\in V,W_a\in W$$

(5)

$$\sum _{a=1}^3\sum _{b=1}^3{y}_{iab}\le 1,v_i\in V,W_a,W_b\in W$$

(6)

$$\sum _{j\in \left\{j|v_j\in B(i)\right\}}{x}_{ijb}+\sum _{j\in \left\{j|v_j\in A(i)\right\}}{x}_{jia}\ge 2y_{iab},v_i\in V,W_a,W_b\in W$$

(7)

$$\sum _{j\in \left\{j|v_j\in B(i)\right\}}\sum _{a=1}^3{x}_{ija}-\sum _{j\in \left\{j|v_j\in A(i)\right\}}\sum _{a=1}^3{x}_{jia}=\left\{\begin{array}{c}\begin{array}{cc}1,& i=1\end{array}\\ 0,i\ne 1,n,v_i\in V\\ -\begin{array}{cc}1,& i=\mathrm{n}\end{array}\end{array}\right.$$

(8)

$$\sum _{a=1}^3\sum _{i=1}^{\mathrm{n}}\sum _{j=1}^n{x}_{ija}{t}_{ija}+\sum _{a=1}^3\sum _{b=1}^3\sum _{j=1}^n{x}_{iab}{t}_{iab}\le T,v_i,v_j\in V,W_a,W_b\in W$$

(9)

The significant weight characteristics of oversize and heavy cargoes necessitate modifications to existing highway and railway bridges due to potential inadequacy in load-bearing capacity. Additionally, vehicle profiles may exceed route clearances, and transshipment capacity at nodes may be insufficient during multimodal transportation. Consequently, bridge modifications are required to adhere to the following constraints:

Equation (2) mandates that the combined weight of the vehicle and cargo post-loading must not surpass the bearing capacity of the reconstructed bridge. Equation (3) articulates that the aggregate profile of the laden vehicle and cargo must remain below the route clearance post-modification. Equation (4) dictates that if transshipment occurs at nodes, the cargo must adhere to the transshipment capacity of the nodes. Equation (5) specifies that transportation between adjacent nodes (i.e., on an arc) within the model must exclusively utilize one of the modes of transportation: road, rail, or waterway. Equation (6) permits transshipment at nodes but restricts the number of transshipments to once per node. Equation (7) encompasses transportation continuity. In the OMTP-OHC, the decision-making process persists until the destination sequence is reached upon encountering a transshipment node, ensuring transportation continuity. Equation (8) concerns node input–output equilibrium. In the decision-making process for the multimodal transportation of oversize and heavy cargoes, the origin city functions as the input point, the destination city as the output point, and all other nodes serve as intermediate nodes, maintaining input–output equilibrium for intermediate nodes. Equation (9) imposes a total time constraint. Given that OHC typically comprises crucial project equipment, any delivery delay results in substantial costs. Hence, the total time must not exceed the carrier’s longest expected delivery time.

3.2. The Modal under the ETS Policy Scenario

Under the ETS policy scenario, enterprises must consider the impact of the emission trading price on cost and path optimization to achieve their goal of profit maximization. Therefore, this chapter establishes an OMTP-OHC model under the ETS policy ($M2$), taking into account the emission trading price.

Based on the above considerations, taking the lowest transportation cost as the optimization objective, the OMTP-OHC modal under the ETS policy is constructed as follows:

$$\begin{gathered} \mathrm{m}\mathrm{i}\mathrm{n}{Z}_2=\sum _{a=1}^3\sum _{i=1}^n\sum _{j=1}^n{x}_{ija}\left(c_{ija}^1+c_{ija}^2\right)+\sum _{a=1}^3\sum _{b=1}^3\sum _{i=1}^n{y}_{iab}\left(c_{iab}^1+c_{iab}^2\right) \\ +(\sum _{a=1}^3\sum _{i=1}^n\sum _{j=1}^n{Q}_a{e}_{ija}{d}_{ija}{x}_{ija}+\sum _{a=1}^3\sum _{b=1}^3\sum _{i=1}^n{Q}_a{e}_{iab}{y}_{iab}-E_p) \end{gathered}$$

(10)

which is also subject to the following conditions when satisfying constraints (2)~(9):

$$\begin{gathered} \sum _{a=1}^3\sum _{i=1}^n\sum _{j=1}^n{Q}_a{e}_{ija}{d}_{ija}{x}_{ija}+\sum _{a=1}^3\sum _{b=1}^3\sum _{i=1}^n{Q}_a{e}_{iab}{y}_{iab}=E_p+S, \\ {v}_i,v_j\in V,W_a,W_b\in W \end{gathered}$$

(11)

$$C_E~N(\mu ,{\sigma }^2)$$

(12)

The objective function $Z_2$ aims to minimize the total cost under the ETS policy, comprising three components. The first two parts of this costing are the same as those in Section 3.1, Ordinary Modal. The third component encompasses the carbon quota trading costs generated throughout the entire process.

Equation (11) represents the carbon emission quota constraint, ensuring that the difference between the carbon emission quota and the total carbon emissions from transportation equals the carbon trading value on the market. Here, a negative carbon trading value ($S$) indicates selling one’s own carbon emission quota for profit, while a positive $S$ implies purchasing carbon emission quotas on the market, incurring costs. Equation (12) imposes constraints on the range of emission trading prices ($C_e$). Based on fitting historical data, it can be assumed to follow a normal distribution within the specified interval.

To clarify, the carbon quotas considered in this paper are not the total quotas allocated to enterprises by relevant departments, but rather the preset carbon quota allocated to the project in the OMTP-OHC model under the carbon trading policy by the enterprises themselves.

3.3. The Modal under the Carbon Tax Policy Scenario

Carbon tax is a policy for pricing carbon emissions, levying taxes on carbon dioxide emissions to incentivize economic entities to adopt energy-saving and emission-reducing measures. By internalizing carbon emissions as costs, it aims to achieve carbon emission internalization. The OMTP-OHC model under the carbon tax policy ($M3$) is developed based on considering the influence of a carbon tax on transportation costs and carbon emissions.

The $M3$ model aims to minimize total costs while considering constraints such as transportation time, loading capacity, and traffic capacity. The formulated decision-making model for transportation routes of oversize and heavyweight cargoes under the carbon tax policy is presented as follows:

$$\begin{gathered} \mathrm{m}\mathrm{i}\mathrm{n}{Z}_3=\sum _{a=1}^3\sum _{i=1}^n\sum _{j=1}^n{x}_{ija}\left(c_{ija}^1+c_{ija}^2\right)+\sum _{a=1}^3\sum _{b=1}^3\sum _{i=1}^n{y}_{iab}\left(c_{iab}^1+c_{iab}^2\right) \\ +(\sum _{a=1}^3\sum _{i=1}^n\sum _{j=1}^n{Q}_a{e}_{ija}{d}_{ija}{x}_{ija}+\sum _{a=1}^3\sum _{b=1}^3\sum _{i=1}^n{Q}_a{e}_{iab}{y}_{iab})\times C_{tax} \end{gathered}$$

(13)

which is subject to Equations (2)–(9).

Objective function (13) is structured into three components in the carbon tax model, which also aims to minimize the total cost. The first two parts of this costing align with those described in Section 3.1, Ordinary Modal. The third component encompasses the carbon quota trading costs generated throughout the entire process. The third component relates to the carbon tax expenditure, which is derived from the carbon emissions generated during the entire transportation process.

4. Solution Approach

4.1. Framework for a Solving Mechanism

The initial population is generated using topological sorting, which ensures that each chromosome adheres to the transportation network structure requirements. Furthermore, randomness is introduced when forming the initial population to enhance the richness and diversity of gene segments, thus preventing the creation of illegal chromosomes.

In the study of the OMPT-OHC under various carbon pricing policies, three distinct models were constructed in Section 3. Given the similarities in objective functions, constraints, and the fact that all models pertain to multimodal transportation path-optimization problems with analogous solution methods, a genetic algorithm was selected for their implementation, tailored to the problem’s specific characteristics. This approach was chosen due to the models’ categorization under the same problem type, which involves multiple transitions and complex intermediate nodes. When the number of nodes is large or the problem size is substantial, a combinatorial explosion can easily occur with an exact solution approach, making it challenging to directly obtain an exact solution. Therefore, this problem is commonly addressed using the genetic algorithm, a type of heuristic algorithm. Using the genetic algorithm enhances the global search capability when solving such problems.

Traditional genetic algorithms often face issues like premature convergence and becoming trapped in local optima. Incorporating the catastrophe operator and elite retention strategy aims to enhance the algorithm’s global search capability and stability. Figure 1 illustrates the specific procedure of the genetic algorithm.

4.2. The Design of the Enhanced Genetic Algorithm

4.2.1. Encoding and Decoding

The formulation of decision-making solutions for the OMTP-OHC requires consideration of two key components: the order of nodes and the corresponding transportation modes between successive nodes. To address this, we design a two-dimensional encoding scheme, where each entry represents a city node and its associated transportation mode for the segment between two consecutive nodes.

In the multimodal transportation network, there are a total of n nodes, where d = [$d_1$ $d_2\dots d_t\dots d_n$] represents a permutation of the network, with $d_1$ = $v_s$, and $d_k$ = $v_t$.

$D=[d_1$ $d_2\dots d_m$] represents a sequence of effective nodes in the multimodal transportation network from the origin to the destination, while $[d_m \dots d_n$] denotes redundant nodes in the entire node set that are not involved in transportation. R = [$R_1$ $R_2\dots R_m\dots R_n$] represents a set of transportation modes, where [$R_1$ $R_2\dots R_{m-1}$] denotes the transportation modes between effective nodes, and $R_k=W_a$ indicates the adoption of mode $W_i$ between $d_k$ and $d_{k+1}$ ($W_a$ represents a transportation mode, where $a=\mathrm{1,2},3$, with $W_1$ denoting road, $W_2$ denoting railway, and $W_3$ denoting waterway). $[R_m \dots { R}_n$] constitutes a sequence of invalid transportation modes between redundant nodes, all set to 0. The purpose of setting redundant nodes $[d_m \dots d_n$] and invalid transportation modes $[R_m \dots { R}_n$] is to maintain a consistent length for each chromosome, thereby reducing the error rate during the algorithm iteration process. The resulting two-dimensional sequence is as follows:

$$\left[\begin{array}{cc}{d}_1& d_2\\ R_1& R_2\end{array} \begin{array}{ccc}...& d_{m-1}& d_m\\ \dots & R_{m-1}& R_m\end{array}\begin{array}{ccc}{ d}_{m+1}& ...& d_n\\ R_{m+1}& \dots & R_n\end{array}\right]$$

For example, suppose there are 8 nodes, with node 1 as the starting point and node 8 as the endpoint. A randomly generated sequence might resemble the one depicted in Figure 2:

4.2.2. Constructing the Initial Population and Fitness Function

1

Construct the initial population

The initial population is generated using topological sorting, which ensures that each chromosome adheres to the transportation network structure requirements. Furthermore, randomness is introduced when forming the initial population to enhance the richness and diversity of gene segments, thus preventing the creation of illegal chromosomes.

The process is as follows: First, the corresponding starting node from the network graph is selected as the first node. Next, based on the network graph’s connectivity, a transportation node adjacent to the previous one is randomly chosen as the next node, continuing this selection until a complete transportation sequence is formed. Subsequently, the first two steps are repeated until the initial population reaches the desired size. In this manner, each individual in the initial population represents a feasible solution that adheres to the network graph’s structure and the transportation mode’s constraints.

2

Construct the fitness function

Considering the need to maximize the fitness value, the fitness function related to the comprehensive objective function is constructed using Equation (14):

$$Fit\left(i\right)=1-{\displaystyle \frac{A_i}{\sum _{j=1}^m{A}_j}}$$

(14)

where $A_i$ represents the comprehensive objective function value of the i-th individual, m is the population size (i.e., the number of individuals in the population), and $\sum _{j=1}^m{A}_j$ denotes the sum of the comprehensive objective function values of all individuals in the population.

4.2.3. Selection, Crossover, and Mutation Operations

1

Selection

Individuals are selected from the population based on their fitness to establish the next generation of the parent population using the roulette wheel selection method. The selection probability for each individual is assigned according to its fitness value. To prevent premature convergence to local optima, the chromosomes’ transportation paths and mode sequences are concatenated into a single string. Duplicate individuals with the same phenotype in both parent and offspring populations are removed. Subsequently, a certain number of individuals are selected as the new parent population based on their fitness values.

2

Crossover

First, according to the crossover probability, select the chromosomes $F1$ and $F2$ from two parent individuals. Determine if there are common path nodes between the selected $F1$ and $F2$ that lie between the starting point $v_s$ and the end point $v_t$. Second, if such common path nodes exist, randomly select one common node from the transportation path node sequences of $F1$ and $F2$. Third, if the position of this common node in the path sequence is $v_i$, then cut the chromosomes at vi and exchange the transportation path node sequences from $v_{i+1}$ to $v_t$ between the two chromosomes. This results in two offspring with valid sequences of transportation path nodes. The transportation mode sequences are then crossed over at the position immediately before the crossover point in the parent’s transportation node sequence (i.e., at position $v_{i-1}$), yielding two offspring transportation mode sequences. The crossover operation is illustrated in Figure 3.

Finally, based on the full permutation $V$, generate the remaining invalid sequences randomly using the intermediate nodes not involved in the transportation. If no common nodes exist between the starting and ending points of the transportation node sequences of both parent chromosomes, reselect parent individuals for a secondary crossover operation. Chromosomes that do not share common nodes between the starting and ending points with any other parent chromosome will not undergo crossover operations.

3

Mutation

During mutation, only the transportation mode sequence undergoes mutation. According to the mutation probability Pm, a mutant individual is selected, and a non-zero gene is randomly chosen in the transportation mode sequence. Then, one of the three transportation modes existing between nodes is randomly selected and replaced with a different transportation mode to form a new transportation mode sequence. This process results in the generation of a new mutant offspring. The mutation operation is depicted in Figure 4.

4.2.4. Elite Retention Strategy and Catastrophe Operation

1

Elite retention strategy

During the crossover and mutation process to generate new offspring, there is a risk of losing the best individual obtained so far. To prevent the loss of the optimal solution, an elite retention strategy is employed. In the selection of offspring, the current best individual is directly replicated to ensure the preservation of the historically best individual obtained by the algorithm.

2

Catastrophe operation

Introducing a catastrophe operator in genetic algorithms can help the algorithm escape local optima and enhance the diversity of the population. The catastrophe operator is typically triggered when the algorithm stagnates, indicated by no significant change in the best solution over several consecutive generations. Define the mutation trigger condition: if the optimal solution does not improve for consecutive $X$ generations (Catastrophe Trigger Generation), activate the mutation. Execute the mutation operation: in this process, disrupt all individuals in the current population except the optimal individual, and generate new individuals using the method to construct initial feasible solutions. Resume algorithm operation: after the mutation operation, the algorithm resumes its selection, crossover, and mutation processes. The updated population facilitates the exploration of new solution spaces.

4.2.5. The Termination Condition of the Algorithm

If the best solution does not change for the maximum consecutive generations $N$, or when the maximum number of generations $G$ is reached, the evolutionary process is terminated. The current best individual is then decoded and output.

4.3. Detailed Steps of the Enhanced Genetic Algorithm

The proposed enhanced genetic algorithm designed in this paper is as follows:

Input: information of OHC, the maximum consecutive generations $N$, the maximum number of generations $G$, Catastrophe Trigger Generation $X$.

Output: optimal path scheme.

Step 1: Import the basic data of the OHC joint transportation network.

Step 2: Generate a feasible initial population randomly based on the topological sorting rule, calculate its fitness, and reset the mutation counter to zero.

Step 3: Perform crossover and mutation on the population, and sort the offspring by fitness after mutation.

Step 4: Select the new parents from the current mutated population and the parent population based on their fitness.

Step 5: Check the termination conditions for the new parent population. If the conditions are met, decode the chromosomes, generate the best individual, and output the result. If the conditions are not met, proceed to the next step.

Step 6: Check whether there is a difference in fitness values between the new parent population and the previous generation of parent population. If there is a difference, reset the catastrophe counter to zero and proceed to Step 7. If there is no difference, increment the catastrophe counter by one and proceed to Step 8.

Step 7: Perform the elitism operation by retaining the individual with the highest fitness in the new parent population. This ensures that the best individual during evolution is directly copied to the offspring without mating. Then, return to Step 3.

Step 8: Check whether the catastrophe counter has reached the catastrophe condition. If the condition is met, proceed to Step 9; otherwise, go back to Step 7.

Step 9: Perform a catastrophe, regenerate all individuals in the population except for the optimal individual, and then return to Step 3.

5. Case Study

5.1. Testing Framework and Algorithm Validation

5.1.1. Testing Instances and Parameters

A shipper demands that a transportation enterprise transport an LY low-end converter transformer which weighs $330$ t and has external dimensions of $9720$ mm × $3450$ mm × $4695$ mm from the starting point ($=1$) to the destination ($=25$) within a limited time of 15 days. Its transportation network is shown in Figure 5, where the numbers represent the numbers of the nodes, and if there is a connecting line segment between two nodes it means that there is at least one mode of transportation available between the two nodes.

The data for this study were sourced from China Special Article Logistics Co., located in Beijing, China. Utilizing existing data on the transportation of out-of-gauge freight across Chinese rail-roads, and integrating highway and waterway network data, we established a multi-modal transportation network graph for LY converter transformers. This network is depicted in Figure 5. This graph includes information on the distances between nodes across three different modes of transportation. Expert consultations revealed that the transportation costs per kilometer for road, rail, and waterway are approximately USD 3743.95, USD 1893.01, and USD 1402.23, respectively. According to the literature [39], the carbon emission coefficients for road, railroad, and waterway transportation in cargo movements are 0.092 kgCO₂/t km, 0.08 kgCO₂/t km, and 0.08 kgCO₂/t km, respectively. The average transit cost, time, and carbon emission when changing between different modes of transportation were obtained by combing them separately and the data are shown in

Table 2. Transshipment cost (USD), time (h) and carbon emission (t).

Cost/Time/Carbon Emission	Highway	Railway	Waterway
Highway	0/0/0	11,218/7/0.128	22,436/15/0.113
Railway	11,218/7/0.128	0/0/0	22,436/16/0.117
Waterway	22,436/15/0.113	22,436/16/0.117	0/0/0

[40].

Due to several constraints, such as the insufficient capacity of transportation nodes, vehicles transporting transformers are precluded from traversing specific routes or nodes.

Table 3. Reconstruction costs of lines. Note: H—Highway; R—Railway.

Node1	Node2	Mode	Cost (USD)	Node1	Node2	Mode	Cost (USD)	Node1	Node2	Mode	Cost (USD)
1	2	R	7011	8	14	R	21,033	15	24	R	14,022
1	3	H	7011	9	14	R	14,022	16	25	R	8413
2	4	R	7011	9	10	R	14,022	17	19	H	7011
2	5	H	14,022	10	15	H	7011	17	19	R	7011
2	5	R	22,436	10	15	R	4207	17	24	R	7011
2	6	R	7011	11	12	R	21,033	18	19	H	16,827
4	6	H	14,022	11	16	H	14,022	18	19	R	7011
4	6	R	7011	11	16	R	28,045	18	20	H	7011
4	8	R	7011	12	16	H	5609	19	22	H	7011
4	9	R	4207	12	16	R	2804	19	22	R	21,033
5	11	H	14,022	13	14	R	7011	19	24	H	14,022
5	11	R	18,229	13	17	H	7011	19	24	R	7011
6	12	R	8413	13	17	R	7011	22	23	R	7011
6	10	R	4207	14	15	H	14,022	23	25	R	14,022
7	8	R	7011	14	15	R	14,022	24	25	H	14,022
8	13	H	7011	15	16	H	16,827	24	25	R	7011
8	13	R	7011	15	16	R	9816	3	4	R	7011

and

Table 4. Reconstruction costs of nodes (USD). Note: A—All modes of transportation are interchangeable; H-W: Transshipment between highway and waterway.

Node	Mode	Cost
7	H-R, R-W	701.11
8	R-W	420.67
14	A	701.11
18	R-W, H-W	560.89
20	H-R, H-W	560.89
23	A	701.11

detail those routes and nodes that are currently impassable but could be modified to ensure accessibility. This modification would lead to both carbon emissions and reconstruction costs. The carbon emissions from these reconstructions are quantified as 0.358 t/m, as referenced in [41], and the cost associated with the reconstruction is estimated at USD 1402.23 for every 7 m. The associated costs are also documented in Table 3 and Table 4.

To analyze the fluctuation characteristics of China’s current ETS prices, this study collects data on emission trading unit prices from the Tianjin Carbon Emissions Trading website. These data reflect the average prices of carbon quota listing agreement transactions in pilot carbon trading markets nationwide, covering the period from 4 January 2022 to 4 April 2023, with a total of 302 entries. The Jarque–Bera test, kurtosis and skewness test, and QQ plot were used to determine if the fluctuations in unit price data followed a normal distribution.

Table 5. The normality test of current price data for ETS.

Inspection Method	Entry	Output
Jarque–Bera test	Average Transaction Price Data	H = 0, p = 0.3338
Skewness–Kurtosis test	As above	Skewness: −0.086718 Kurtosis: 2.6439

presents the results of the Jarque–Bera test and the Skewness–Kurtosis test, indicating that the ETS unit price data conform to a normal distribution. Specifically, the mean value is USD 8.083, and the variance is 0.03. These tests are essential for validating the assumption of normality, which is crucial for subsequent statistical analyses. The QQ plot shown in Figure 6 further corroborates this finding by visually confirming the normality of the data distribution. For simplicity in calculations and to maintain consistency in reporting, we round the mean and variance to two decimal places, resulting in USD 8.084 and 0.03, respectively.

The algorithm parameters are set as follows: the crossover rate is 1.0, the mutation rate is 0.9, the population size (m) is 40, and a catastrophe event occurs when the generation count reaches 20($X$). The maximum number of iterations ($G$) is 120, and the algorithm can terminate if the same optimal solution is achieved for 80 consecutive generations ($N$). The algorithm is implemented using MATLAB R2016a.

5.1.2. Algorithm Validity Testing

In the ETS policy model, both the traditional genetic algorithm (GA) and the improved genetic algorithm proposed in this paper are used for optimization tests. The optimization results are shown in Figure 7 and Figure 8, where the horizontal axis represents the number of iterations, and the vertical axis represents the lowest cost obtained by the current population (cost unit: million dollars).

Comparing the results of the two algorithms, it can be seen that the traditional GA stabilizes at an optimal cost of USD 331,487.06 after 28 iterations. In contrast, the improved GA reaches the same cost level more quickly, after just 20 iterations. Notably, the im-proved GA escapes the local optimum through the catastrophe mechanism at the 40th iteration, obtaining a new optimal solution with a cost of USD 329,397.74. Furthermore, it achieves an even better solution at the 51st generation, with a total cost of USD 329,012.13. These results demonstrate that the improved GA has superior global search capability, providing the potential for finding better solutions.

5.2. Analysis of Results in the Ordinary Scenario

In this section, we obtain the optimal route scheme for the transformer’s multimodal transportation without the influence of any carbon pricing policy and with only the single objective of minimizing the cost.

Table 6. The lowest-cost solution set under the ordinary scenario.

Path Numbers	Path Schemes	Cost/USD	Carbon Emissions/t	Time/h	Number of Shipping Mode Changes	Mileage/ km
A1	$1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$5\stackrel{R}{\to }$$11\stackrel{R}{\to }$$16\stackrel{R}{\to }$25	329,012.13	161.72	47.36	0	1301
A2	$1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$5\stackrel{R}{\to }$$11\stackrel{W}{\to }$$16\stackrel{R}{\to }$25	329,397.74	113.19	150.03	2	2828
A3	$1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$6\stackrel{R}{\to }$$10\stackrel{R}{\to }$$15\stackrel{R}{\to }$$16\stackrel{R}{\to }$25	331,487.06	81.53	55.23	0	1582
A4	$1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$6\stackrel{R}{\to }$$12\stackrel{R}{\to }$$16\stackrel{R}{\to }$25	331,725.44	68.60	57.13	0	3996
A5	$1\stackrel{R}{\to }$$3\stackrel{R}{\to }$$4\stackrel{R}{\to }$$9\stackrel{W}{\to }$$15\stackrel{W}{\to }$$16\stackrel{R}{\to }$25	333,506.28	48.41	200.53	2	2692
A6	$1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$6\stackrel{R}{\to }$$12\stackrel{W}{\to }$$16\stackrel{R}{\to }$25	333,618.45	65.93136	124.24	2	2846

presents the top six ranked path sets based on fitness.

The results in Table 6 indicate that all six path schemes have selected railway transportation and rail–water multimodal transport. Notably, no road transportation appears in any of the paths. Thus, it can be inferred that in cost-optimized decisions for multimodal OHC transportation, railway and waterway transportation are more frequently utilized, while road transportation has a relatively weaker advantage.

A detailed analysis of the top six cost-optimal path schemes reveals that the lowest cost scheme is Path A1, which involves railway transportation and incurs a cost of USD 329,012.13. Among these paths, Path A5 has the smallest carbon emissions: it utilizes rail–water multimodal transport, emitting 48.41 tons of carbon. From an economic perspective, Path A1 has the advantage of being the most cost-effective and having the shortest travel time. Railway transportation also minimizes risk. However, Path A1’s carbon emissions are 234.06% higher than those of Path 5, which is not aligned with national energy-saving and emission-reduction goals. Path A5, although slightly more expensive than Path A1 by USD 4494.15, has the lowest carbon emissions. It aligns with the concept of low-carbon transportation. However, Path A5’s longer travel time (153.17 h) and higher number of transitions, with a significant proportion of waterway transport, may pose some risk of cargo damage.

5.3. Analysis of Decision-Making under Actual ETS Policy

5.3.1. The Impact of Fluctuations

Using the ETS Model (M2), with parameters of a carbon quota of 100 tons and an emission trading price of USD 8.08 per ton, the algorithm yields the top six optimal solutions, as recorded in

Table 7. The lowest-cost solution set under an ETS policy with an emission trading price of ¥57.64 per ton.

Path Numbers	Path Schemes	Cost/USD	Carbon Emissions/t	Time/h	Number of Shipping Mode Changes	Mileage/ km
B1	$1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$5\stackrel{R}{\to }$$11\stackrel{R}{\to }$$16\stackrel{R}{\to }$25	329,511.04	161.7237	47.36	0	2828
B2	$1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$5\stackrel{R}{\to }$$11\stackrel{W}{\to }$$16\stackrel{R}{\to }$25	329,497.30	113.1900	150.03	2	1301
B3	$1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$6\stackrel{R}{\to }$$10\stackrel{R}{\to }$$15\stackrel{R}{\to }$$16\stackrel{R}{\to }$25	331,393.96	81.53	55.23	0	1582
B4	$1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$6\stackrel{R}{\to }$$12\stackrel{R}{\to }$$16\stackrel{R}{\to }$25	331,471.64	68.6000	57.13	0	3996
B5	$1\stackrel{R}{\to }$$3\stackrel{R}{\to }$$4\stackrel{R}{\to }$$9\stackrel{W}{\to }$$15\stackrel{W}{\to }$$16\stackrel{R}{\to }$25	333,082.24	48.41	200.53	2	2692
B6	$1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$6\stackrel{R}{\to }$$10\stackrel{R}{\to }$$15\stackrel{W}{\to }$$16\stackrel{R}{\to }$25	333,343.06	65.93	124.24	2	2846

. Given that the real ETS price is market-based and fluctuates with market conditions, it exhibits inherent stochasticity. Therefore, to enhance the realism of the model, we simulate the price fluctuations using a normal distribution based on actual market prices and generated 1000 price samples that conform to this normal distribution to reflect the realistic market conditions. Subsequently, we calculated the proportion of optimal solutions corresponding to paths B1 to B6 in Set 2. The results, illustrated in Figure 9, show the distribution of optimal solutions across these paths.

Based on Figure 9, it can be observed that under the current ETS policy, price fluctuations have a relatively limited impact on path selection. Specifically, in the simulation, Path B2 becomes the optimal choice with an approximately 91% probability, while Path B1 is selected with a 9% probability. This result indicates that under the current Chinese ETS price, despite fluctuations in the emission trading price, its impact on path decisions with the objective of cost minimization is relatively weak. While the current ETS price effectively encourages more decarbonized decisions with a high probability, there remains a small chance that the decision scenario will be the same as under a no-carbon emission policy.

Although the impact of emission trading price fluctuations on total costs is relatively small, such fluctuations may influence carriers’ path decisions in the long run. Carriers may need to consider strategies to reduce their sensitivity to emission trading price fluctuations. Furthermore, this figure has implications for policymakers, as it demonstrates that monitoring and controlling the market to maintain the ETS price within a rational range can incentivize firms to choose more low-carbon options.

5.3.2. Sensitivity Analysis of Carbon Quotas

Carbon quotas are the allowable emissions of carbon dioxide set by the government to encourage companies to reduce energy consumption and emissions. When formulating transportation strategies, companies need to consider not only traditional cost factors such as transport costs, transshipment costs, and reconstruction costs, but also the costs associated with allocating carbon quotas. If carbon quotas are insufficient, companies may need to purchase additional quotas in the market, thereby increasing the total cost. Conversely, if quotas are abundant, companies can generate revenue by selling excess quotas.

Therefore, the prediction and allocation of carbon quotas by enterprises may affect the OMTP-OHC. By varying the quantity of carbon quotas, we can analyze carriers’ decisions under different carbon quota scenarios and thus understand the impact of changes in carbon quotas on decision-making.

Since the normal distribution has the highest probability of occurrence at its mean value, we can utilize the 68-95-99.7 rule to understand its characteristics. According to this rule, for any normal distribution, approximately 68.3% of the data fall within the interval of the mean ± 1 standard deviation, 95% of the data fall within the mean ± 2 standard deviations, and 99.7% of the data fall within the mean ± 3 standard deviations. Therefore, the price of ETS has a high probability of fluctuating between 7.67 and 8.48.

From

Table 8. Optimal pathway choices under different carbon quotas.

	Carbon Price (USD/t)	7.67	7.79	7.91	8.15	8.23	8.32	8.40	8.48
Carbon Quotas (t)
0		B1	B1	B2	B2	B2	B2	B2	B2
20		B1	B1	B2	B2	B2	B2	B2	B2
40		B1	B1	B2	B2	B2	B2	B2	B2
60		B1	B1	B2	B2	B2	B2	B2	B2
80		B1	B1	B2	B2	B2	B2	B2	B2
100		B1	B1	B2	B2	B2	B2	B2	B2
120		B1	B1	B2	B2	B2	B2	B2	B2

B1 denotes scheme B1 in Table 6.

, we observe that under constant carbon prices, increasing carbon quotas leads to a slight increase in total costs, although this increase is relatively small. Interestingly, variations in carbon quota levels at the same carbon price do not significantly alter the optimal path. This indicates that in a stable emission trading price market under ETS policy, total costs are not highly sensitive to changes in carbon quotas.

When the emission trading price is between USD 7.67/t and USD 7.79/t, the optimal path remains unchanged (Path B1) despite changes in carbon quotas. This stability suggests that companies can increase carbon quotas without significantly affecting costs. However, when the price ranges from USD 7.91/t to USD 8.48/t, the optimal path shifts to Path B2, indicating a preference for more energy-efficient routes at higher emission trading prices. This further illustrates that at higher emission trading prices, carriers tend to choose more energy-efficient and emission-reducing routes.

Overall, the impact of carbon quotas within a reasonable range on total costs is limited, showing that carriers are not highly sensitive to these quotas under the current ETS mechanism.

The potential reasons for the low sensitivity of the OMTP-OHC model to ETS price fluctuations and preset carbon quotas may be as follows: On the one hand, this may be due to the characteristics of the cargo and the composition of multimodal transportation costs: the transportation and modification costs of OHC are significantly higher than the carbon emission costs. Since these costs dominate the total expenses, the impact of carbon costs under the current Chinese ETS policy on the overall cost structure is limited. On the other hand, the current ETS policy prices in China are not considered high, and even seem to be slightly on the lower side. In the model constructed in this paper, price fluctuations are simulated within a reasonable range based on market data. Due to the slightly low carbon prices, the resulting carbon emission costs are also low. In the case studies used in this paper, carbon trading costs account for only a small portion of the total costs, and thus may have a limited impact on the overall costs.

For enterprises, setting a reasonable carbon quota for a given transport task within the estimated range has a relatively small effect on overall costs. Conversely, changes in ETS prices significantly influence route decisions, and raising the ETS price can incentivize companies to adopt more proactive emission reduction measures.

5.4. Analysis of Decisions under Carbon Tax Policy

Based on the current global carbon pricing, this paper categorizes carbon taxes into five levels: low carbon tax (USD 3.65/t), lower-intermediate carbon tax (USD 8.40/t), intermediate carbon tax (USD 50.48/t), upper-intermediate carbon tax (USD 91.14/t), and high carbon tax (USD 131.81/t). This classification aims to analyze the impact of different carbon tax levels on the OMTP-OHC. Using these five carbon tax prices as input parameters for the carbon tax model, we calculate the optimal decision paths under each tax level. The optimal solutions obtained under different carbon tax levels are presented in

Table 9. Optimal path scenarios under different carbon tax levels.

Carbon Tax Levels	Transportation Scheme	Cost/USD	Carbon Emissions/t
Low	$\mathrm{C}1:1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$5\stackrel{R}{\to }$$11\stackrel{R}{\to }$$16\stackrel{R}{\to }$25	329,601.07	161.72
Lower intermediate	$\mathrm{C}2:1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$5\stackrel{R}{\to }$$11\stackrel{W}{\to }$$16\stackrel{R}{\to }$25	330,344.95	113.19
Intermediate carbon tax	$\mathrm{C}3:1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$6\stackrel{R}{\to }$$10\stackrel{R}{\to }$$15\stackrel{R}{\to }$$16\stackrel{R}{\to }$25	33,5115.75	81.53
Upper-intermediate level	$\mathrm{C}4:1\stackrel{R}{\to }$$3\stackrel{R}{\to }$$4\stackrel{R}{\to }$$9\stackrel{W}{\to }$$15\stackrel{W}{\to }$$16\stackrel{R}{\to }$25	337,931.85	48.41
High	$\mathrm{C}4:1\stackrel{R}{\to }$$3\stackrel{R}{\to }$$4\stackrel{R}{\to }$$9\stackrel{W}{\to }$$15\stackrel{W}{\to }$$16\stackrel{R}{\to }$25	339,881.51	48.41

.

This section examines the impact of different carbon tax levels on decision-makers’ route choices by comparing carbon tax prices. Under a low carbon tax price of USD 3.65/t, the cost-optimal solution C1 remains the same as the optimal solution A1 under a zero-carbon emission policy. This suggests that at low carbon tax levels, decision-makers are not sufficiently incentivized to shift toward more energy-efficient and low-carbon routes. However, increasing the carbon tax price to USD 8.40/t (a lower-intermediate carbon tax) leads decision-makers to switch from route C1 to route C2. Despite a cost increase of USD 743.88, this change results in a significant 30.01% reduction in carbon emissions.

At an intermediate carbon tax level, emission reduction becomes even more pronounced. Compared to the previous two carbon price levels, the intermediate carbon tax prompts decision-makers to favor route C3, representing a more carbon-efficient transportation path. Carbon emissions decrease by 49.59%, but costs increase by USD 5514.69 compared to route C1. Results obtained under upper intermediate and high carbon tax prices further confirm the significant impact of increasing carbon tax prices on emission reduction. In both cases, decision-makers tend to choose the most energy-efficient and low-carbon route, represented by route C4. However, despite the upper-intermediate carbon tax price, there is no significant change in route selection during the final decision-making process; this suggests that decision sensitivity to carbon tax price variations may diminish in a high-carbon tax environment. Consequently, setting an appropriate carbon tax level can balance carrier interests while achieving climate change mitigation and energy efficiency goals.

In conclusion, setting a reasonable level of carbon tax is crucial for incentivizing the choice of energy-efficient and low-carbon pathways. Policymakers should consider the economic capacity of carriers while ensuring that the carbon tax price level effectively promotes energy transition and the achievement of carbon neutrality goals.

5.5. Comparative Analysis of Different Policies

When analyzing decision-making under different carbon pricing policies, it is important to note the significant differences in carbon prices under the carbon tax policy. To illustrate the impact of carbon tax prices on decision-making, this section uses three levels of carbon tax prices: lower intermediate, intermediate, and upper intermediate. These levels reflect the different carbon price scenarios discussed in the previous section. The following section comprehensively presents the optimal solutions under different scenarios in

Table 10. Optimal path options for different scenarios targeted at cost.

	In the Ordinary Scenario	In ETS Policy Scenario	In Carbon Tax Policy Scenario
			Lower-Intermediate Level	Intermediate Level	Upper-Intermediate Level
Scheme	$1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$5\stackrel{R}{\to }$$11\stackrel{R}{\to }$$16\stackrel{R}{\to }$25	$1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$5\stackrel{R}{\to }$$11\stackrel{W}{\to }$$16\stackrel{R}{\to }$25	$1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$5\stackrel{R}{\to }$$11\stackrel{W}{\to }$$16\stackrel{R}{\to }$25	$1\stackrel{R}{\to }$$2\stackrel{R}{\to }$$6\stackrel{R}{\to }$$12\stackrel{R}{\to }$$16\stackrel{R}{\to }$25	$1\stackrel{R}{\to }$$3\stackrel{R}{\to }$$4\stackrel{R}{\to }$$9\stackrel{W}{\to }$$15\stackrel{W}{\to }$$16\stackrel{R}{\to }$25
Cost /USD	329,012.13	329,511.04	330,259.41	335,115.75	339,881.51
Carbon emissions/t	161.72	113.19	113.19	81.53	48.41

.

When analyzing decision scenarios under different carbon pricing policies, we can compare them along two dimensions: cost and carbon emissions. This paper provides a specific comparison of route selection under three policies (the ordinary scenario, emissions trading policy, carbon tax policy):

Ranking of carbon emission reduction intensity:

Upper-intermediate level of carbon tax > intermediate level of carbon tax > lower-intermediate level of carbon tax = ETS policy > the ordinary scenario.

Ranking of total costs:

Upper-intermediate level of carbon tax > intermediate level of carbon tax > lower intermediate-level of carbon tax > ETS policy > the ordinary scenario.

It can be observed that the intensity of carbon emission reduction is related to the increase in carbon emission costs. The carbon tax policy, by directly increasing the cost of carbon emissions, strongly encourages decision-makers to choose low-carbon pathways. On the other hand, the ETS policy encourages the selection of low-carbon pathways through market mechanisms, but its emission reduction effect is relatively weak under current prices.

In summary, the carbon tax policy has a more significant effect on emission reduction, but attention should also be paid to preventing excessive carbon taxes from imposing too much cost burden on carriers. Meanwhile, under current ETS policy prices, although the emission reduction effect is weaker, market mechanisms can still promote the selection of low-carbon pathways to some extent. Policymakers must carefully weigh the costs to businesses against the effectiveness of emission reduction when contemplating the implementation or modification of these policies.

6. Conclusions

Based on the comprehensive results of the analysis of this study, several conclusions can be drawn:

Firstly, the enhanced genetic algorithm with an elite strategy and catastrophe mutation proposed in this paper demonstrates exceptional global search capabilities. It effectively identifies multimodal transportation solutions that better meet the optimization objectives of the OMTP-OHC.

Secondly, under the current Chinese ETS policy, the impact of emission trading price fluctuations on total costs is relatively minor. Under fluctuating conditions, the current emission trading prices are likely to steer decision-making towards low-carbon solutions. Additionally, sensitivity analyses of carbon quotas indicate that the total cost of transporting OHC for an enterprise is less sensitive to variations in carbon quotas when emission trading prices are stable. This provides flexibility for enterprises to allocate carbon emission quotas in advance. From the perspective of carbon emissions reduction, under the current ETS pricing, optimizing the OMTP-OHC problem with cost minimization as the decision criterion can lead to route choices with 30% lower carbon emissions compared to the normal scenario. However, under fluctuating conditions, routes with lower carbon emissions than those selected under a fixed ETS price are not chosen. This reflects the limited adaptiveness of the decision-making process to volatile ETS conditions, emphasizing the need for more robust strategies in dynamic pricing environments.

Thirdly, different levels of carbon tax have varying impacts on decision-makers’ path choices. An upper-intermediate level of carbon tax more effectively promotes the selection of energy-saving and emission-reducing pathways compared to lower levels. The impact of different levels of carbon tax on emissions reduction is significant. Specifically, as the carbon tax increases from low to high levels, the potential for emissions reduction escalates from 0% to 70.07%. However, once a higher level of carbon tax is reached, further increases in the tax rate may not result in additional emissions reductions. This suggests diminishing returns on emissions reduction as the carbon tax approaches the upper limits.

For policymakers, the results emphasize the importance of incentivizing low-carbon transportation choices through the reasonable control of emission trading prices and carbon tax levels. Policymakers should ensure that carbon pricing policies effectively promote energy transition and the achievement of carbon neutrality goals, while also considering affordability for carriers.

7. Discussion and Future Research

In the optimization of intermodal transportation routes for oversized and heavy cargo (OHC), existing models rarely take into account the scenario of carbon pricing policies. In contrast, our model not only integrates these policies but also adapts to fluctuations based on actual ETS trading prices in China, enhancing its practicality and relevance.

In comparison with other multimodal transportation optimization models that consider carbon pricing policies for general cargo, this study finds that OHC exhibits greater stability under the fluctuations of China’s ETS policy, with less variability in decision-making. Moreover, the decision-making performance of OHC under various carbon tax levels is similar to that of regular cargo, promoting low-carbon directions. Notably, under a high carbon tax scenario, similar to with general cargo, a marginal effect is observed, where significant changes in decision-making do not occur. Some comparisons with similar studies are as follows.

Comparison with Reference [28]:

On carbon tax policy: Both oversized and heavy cargo (OHC) and regular goods exhibit similar decision-making behaviors across various carbon tax levels, opting for the lowest carbon emission route when the tax reaches a significant threshold. Notably, OHC achieves the lowest carbon emission route at an upper-intermediate level of USD 91.14/t, beyond which no significant decision changes occur. Reference [28] reaches a similar conclusion, with regular goods finding the lowest emission route at a tax of 500 yuan/t, with no further changes thereafter. Thus, it can be inferred that both OHC and regular goods exhibit marginal effects under high carbon tax scenarios.

In model design: Reference [28] sets the ETS price as a constant, whereas this study adopts a more realistic approach by modeling the ETS price as a variable following a normal distribution. Conclusively, while Reference [28] discusses route changes based on hypothetical market prices (500 yuan/t and 1000 yuan/t), this paper analyzes routes based on actual ETS prices, offering potentially more realistic insights into the effects of ETS policies.

Comparison with Reference [31]:

Reference [31] investigates the impact of ETS price uncertainty on regular goods and observes significant total cost variability. In contrast, this study, despite examining the randomness of ETS prices, finds that the total cost variability for OHC is not pronounced. This disparity may be attributed to differences in how ETS prices are modeled and the proportion of carbon emission costs in the total costs of different types of cargo. For OHC, given its relatively low share of carbon emission costs, even significant fluctuations in ETS prices result in limited total cost variability.

While our study focuses on a case that is not extensive in scale, it provides a valuable foundation for understanding the implications of carbon pricing policies. To further research, we suggest expanding the scope to include larger-scale cases with more nodes, based on the following reasons:

Complexity and realism: In real life, different operational scales can introduce new complexities and interactions. Larger systems may involve more nodes, routes, and decision points, which could significantly impact the optimization results under carbon pricing policies. By analyzing larger-scale cases, the robustness and reliability of the model can be tested in more realistic and complex logistics networks.

Generalizability of results: By expanding the research scope to include larger-scale cases, the findings from case studies can be tested to see if they hold true in a broader range of scenarios. This is particularly beneficial for policymakers and industry stakeholders who may apply these insights in practical settings.

Sensitivity to policy changes: Different systems may respond differently to policy changes. While small-scale systems might show only minor effects, in larger-scale systems, small changes might be amplified due to the scale of operations and the cumulative effect of minor changes. Understanding how different scales respond to changes in carbon tax levels, ETS price dynamics, and carbon quota adjustments can provide deeper insights into the dynamics of carbon pricing impacts.

This expansion will help to more thoroughly analyze how different carbon pricing policies affect the cost optimization of the OMTP-OHC across various scales.