Optimization of Multimodal Transport Paths Considering a Low-Carbon Economy Under Uncertain Demand

Abstract

Aiming at the uncertainty in cargo demand in the transportation process, the multimodal transportation path optimization problem is studied from the perspective of a low-carbon economy, and the robust optimization modeling method is introduced. Firstly, a robust optimization model for multimodal transportation is built using the multimodal transportation path optimization model under demand certainty, and the total transportation cost is then calculated by taking into account not just only the cost of transportation and trans-shipment but, additionally, the price of waiting because of schedule restrictions on trains and airplanes. Secondly, carbon emissions are added into the model as a constraint or cost by converting four different low-carbon policies. Then, the simulated annealing mechanism is introduced to improve the ACO algorithm. Finally, solomon calculus is used for the solution. The outcomes demonstrate that the improved annealing ant colony hybrid algorithm simulation can essentially improve the multimodal transportation path optimization problem with uncertain demand and promote multimodal transportation emission reduction. Among the four carbon emission policies, the mandatory carbon emission policy means are tough, and the greatest impact comes from reducing emissions and using less energy. Energy conservation and emission reduction have the second-best impact, while the three policy tools of carbon taxes, carbon trading and carbon payment are more modest.

1. Introduction

As one of the main problems in supply chain management, path optimization has always been a hot research topic [1]. With the rapid development of the modern economy, the demand for transportation is increasing, and carbon emissions are also increasing. In the second United Nations Global Conference on Sustainable Transportation, the United Nations Secretary-General Antonio Guterres pointed out that “Transportation carbon emissions account for more than a quarter of global greenhouse gas emissions” [2]. As an advanced modern logistics technology, multimodal transport can reduce transportation costs and carbon emissions by utilizing the advantages of different modes of transportation. At the same time, because multimodal transportation consists of multiple modes of transportation, compared with a single mode of transportation, it also needs to consider the transfer between different modes of transportation and the influence of railroad and waterway transportation schedules. This makes it more difficult to choose the transportation path for multimodal transportation than for a single mode of transportation, which has attracted extensive attention from scholars in recent years.

With the in-depth implementation of the Paris Agreement, low-carbon policies such as mandatory carbon emissions, carbon tax, carbon trading and carbon compensation have emerged and become the main measures to control carbon dioxide emissions. In the context of green low-carbon, British scholars Turnpenny J [3] conducted the earliest research on different types of regional carbon emission assessments and predictions; Liverman D M [4] conducted research on the carbon offset mechanism and its governance. And in recent years, more and more scholars have begun to consider the carbon emission problem in intermodal transportation. Low-carbon policies, such as mandatory emissions, carbon tax, carbon trading and carbon offsetting, control carbon emissions or emission costs to achieve the purpose of emission reduction and influence economic decision-making. Different transportation mode selection results will be obtained under different carbon emission control policies; thus, it is necessary to further study the multimodal transportation route selection problem under different policies on the basis of analyzing carbon emissions. In addition to the impact of uncertain demand on multimodal transportation path selection, it will also have an impact on carbon emissions in the transportation process, and different modes of transportation have different unit carbon emissions. Therefore, it is necessary to consider the uncertainty in demand when calculating carbon emissions, based on which transportation modes and routes can be rationally selected.

In summary, this paper starts from the uncertainty in transportation demand, based on the comprehensive consideration of a soft time window for the limitation of transportation mode shifts, establishes a robust optimization model of multimodal transportation paths with the objective of minimizing the total cost of transportation through the scenario analysis method in robust optimization, analyzes the changes in the costs and the carbon emissions under different carbon emission policies and improves the solution of the ACO algorithm, which adopts the simulated annealing mechanism. The developed algorithm is tested and validated by a solomon arithmetic example, and the traditional ant colony algorithm is compared with the hybrid ant colony algorithm. This paper is organized as follows: Section 2 presents a detailed literature review on multimodal transportation, low-carbon path optimization and heuristic algorithms. Section 3 presents the problem description and optimization model. In Section 4, the improved ACO algorithm and the solution procedure are described. Section 5 provides a case study of the algorithm and compares the improved algorithm with the unimproved algorithm. Finally, Section 6 draws conclusions and suggests possible future improvements.

2. Literature Review

This paper focuses on the review of the low-carbon multimodal transportation path optimization problem under uncertain demand, which is reviewed in three main areas: demand uncertainty, low-carbon path optimization and heuristic algorithms.

Since Dantzig and Ramser [5] proposed the vehicle routing problem, the main purpose is to reduce the transportation cost and distance. Yuan B [6] investigated how to find vehicle routes and production schedules of 3D printers mounted on vehicles with a minimum travel cost. Multimodal transportation, as a newly emerging mode of transportation in recent years, and its related optimization has gradually attracted the attention of researchers. Giusti R [7] overviewed the synchronous logistics multimodal transportation problem, and Giusti R [8], in turn, solved the synchronous multimodal transportation supply chain problem through the SYNCHRO-NET method. Crainic T G [9] analyzed the multimodal trans-shipment problem and proposed the MILT for the multi-cycle synchronization of a single commodity. The MILP calculation method for the single-commodity multi-cycle synchronization–trans-shipment problem was proposed. Ceulemans E [10] investigated the theory of relevant decision-makers in the intermodal transportation problem. When constructing the multimodal transportation model, problems such as seasonal demand and sudden replenishment exist in the actual cargo transportation process. Consequently, when developing a transportation scheme, cargo transportation is carried out under uncertainty in demand.

Nowadays, scholars’ studies on multimodal path optimization models under demand uncertainty are as follows: Agachai Sumalee [11] developed a stochastic equilibrium selection model for users facing uncertain demand and extreme weather. Zhang [12] investigated the low-carbon path optimization problem under double uncertainty and designed a Monte Carlo sampling-based catastrophic adaptive genetic algorithm (CA-GA) solution. In addition, uncertain demand, besides affecting multimodal transport path selection, will also have an impact on carbon emissions in the transportation process, and there are different unit carbon emissions for different modes of transportation. Therefore, it is necessary to consider the uncertainty in demand when calculating carbon emissions, based on which transportation modes and routes are reasonably selected. Liu S et al. [13] considered the cold chain container intermodal route optimization problem in a time-varying network of carbon emissions and took the carbon emission limit as a constraint to establish a cold chain intermodal route optimization model. Zhang M et al. [14] investigated the intermodal route optimization problem under the uncertainty in the time window and demand and explored the impact of the carbon tax value on carbon emission. Liu W S [15] studied the green multimodal transportation path under the mixed uncertainty condition optimization problem with uncertain parameters using a fuzzy chance constraint model and a robust optimization model.

Among the methods for solving the above path optimization problems, heuristic algorithms have better results in solving various NP-hard problems. Among them, the simulated annealing algorithm, ant colony algorithm, etc., have received attention from many scholars. Harbaoui Dridi I [16] proposed a multi-vehicle multi-warehouse pickup and delivery problem, which improves the particle swarm algorithm in order to solve a variety of continuous and discrete problems. Oudani M [17] developed a mixed integer model and solved it with a simulated annealing algorithm. Some scholars also used hybrid meta-heuristic algorithms to solve the problem, and more and more studies have shown that hybrid algorithms outperform single algorithms in terms of computational efficiency and results. Moghadam [18] combined an ant colony system and simulated annealing to solve order-divisible VRP. Wang Z [19] proposed a hybrid adaptive rescheduling strategy and combined it with a hybrid adaptive rescheduling strategy for the problem of path optimization in an uncertain environment. They designed an adaptive rescheduling strategy for the path optimization problem in an uncertain environment and developed Non-dominated Sorting Genetic Algorithm II (NSGA-II) by combining population initialization rules and multiple genetic operators.

Table 1. Multimodal transportation path optimization constraints and algorithms.

Author	Particular Year	Reference	Objectives	Restriction	Algorithms
Ghoseiri K	2010	[20]	Minimum total fleet size and distance	Time window constraint	Genetic algorithm
Lei Q	2012	[21]	Minimal total cost	Time window constraint	K-shortest circuit method
Liao Z	2016	[22]	Minimal total cost	Carbon trading and carbon tax	Exact algorithm
Lv X.W	2018	[23]	Minimize total cost, carbon footprint	Hybrid time window constraints	Particle swarm based on simulated annealing
Wang	2019	[24]	Minimal carbon footprint	-	Entropy weight fuzzy analysis
Chen N J	2020	[25]	Minimum total transportation costs and maximum satisfaction	Time window for receipt and time frame for transportation	Ant colony algorithm
Yang N	2020	[26]	Minimal total cost	Time window constraint	K-short-circuit–GA hybrid algorithm
Wu P	2023	[27]	Minimal total cost	Carbon emissions and time window constraints	Adaptive genetic algorithm
Zhu P	2024	[28]	Minimal total cost	Delay time, time window constraints	Genetic algorithm
Y. Liu	2024	[29]	Minimize transportation time, cost and carbon emissions	Time window constraint	PSO-GA hybrid algorithm

organizes some references of constraints related to multimodal transport models.

This research creates an amalgam ant colony method that combines the simulated annealing technique with the ant colony algorithm based on the aforementioned study, which adopts the simulated annealing mechanism to generate new solutions and designs the cooling rule to increase the method’s search efficiency concerning the dynamic temperature decay factor. Meanwhile, this work analyzes data to investigate the sensitivity and efficacy of the algorithm itself in order to test the usefulness of the model. The main contributions of this paper are (1) considering four kinds of low-carbon economic policies, namely, mandatory carbon emission, carbon trading, carbon offset and carbon tax, to establish the demand uncertain multimodal transportation path optimization model with various sustainable regulations and (2) introducing the simulated annealing mechanism to design the simulated annealing ant colony algorithm, which proved to be more effective than other algorithms and has a better solving effect under uncertain demand.

3. Materials and Methods

3.1. Problem Description

Let the multimodal transportation network be denoted by $G=\left\{M,N\right\}$, where $M$ and $N$ are the set of nodes, and a group of forms of transportation, $M=\left\{i,j,k\left|i,j,k=1, 2, \dots , m\right.\right\}$ and $\left|M\left|=m\right.\right.$, $N=\left\{1, 2, \dots ., n\right\}$ and $\left|N\left|=n\right.\right.$. As shown in Figure 1, assuming that a multimodal logistics network consists of transportation nodes and trans-shipment nodes, where three different transportation modes are used, i.e., railroads, roads, air transportation networks and aircraft transportation networks, where different modes of transportation are converted through the trans-shipment nodes, while railroads and airlines have schedule restrictions. Now, there is a collection of products with unclear cargo volume to be transported to the end point D at the starting point O. Considering the impacts of schedules and carbon emissions, we try to find the total cost minimization scheme and the combination of transportation routes under the specific carbon emission policy.

Some of the parameters and meanings of the questions are shown in

Table 2. Parameter descriptions.

Parameter	Meaning
$t_{(i,j,t_i)}^m$	The time $t_i$ taken at time point $m$ from $i$ to $j$, min
${t^{\prime }}_{j(m,n)}$	After $j$ is changed from $m$ to $n$, the most recent departure moment of $n$, min
${t^{\prime }}_{j(m)}$	The moment of arrival of $m$ at $j$ in terms of $m$
${t^{\prime }}_i$	The moment of leaving $i$
$c_{i,j}^m$	The duration of time required to be moved from node $i$ to $j$ through $m$, Yuan
$q$	Demand for goods, kg
$c_i^{m,n}$	Cost of unit transit from $m$ node $n$ to $i$, Yuan
$d_{i,j}^m$	Transportation distance from node $m$ to $n$ via $i$
${\mathrm{e}}_{ij}^m$	Carbon emissions from node $i$ to $j$ for transportation mode $m$, kg
$e_i^{mn}$	Carbon emissions per unit when switching from transportation mode $m$ to $n$ at node $i$
$c_w$	Unit waiting costs for waiting for buses to dispatch
$p_s$	Probability of scenario $S$ occurring
$q^s$	Uncertain demand under Scenario $S$
$C_s$	Feasible solutions of the objective function under scenario $\mathrm{s}$
$C_{s*}$	Scenario $S$ determines the optimal solution of the objective function under
$\gamma$	Maximum regret value allowed under scenario $\mathrm{s}$

and

Table 3. Decision-making variables.

Variant	Instructions
$x_{(i,j,t_i)}^m$	At $t_i$, choosing $m$ from $i$ to $j$ equals 1; otherwise, it equals 0
${\mathrm{y}}_j^{m,n}$	After $j$ transitions from $m$ to $n$, it equals 1; otherwise, it equals 0

:

3.2. Model Building

Due to the erratic need for cargo transportation, the scenario method in robust optimization is used. In this paper, the discrete scenarios with known probabilities show the uncertainty in the parameters. The probability of occurrence of each scenario is determined, and a multimodal transit path strategy under uncertainty is established by using the scenario analysis method in robust optimization as follows:

$$C_1={\displaystyle \sum _{m\in M}{\displaystyle \sum _{i=s}^N{\displaystyle \sum _{j=s}^N{c}_{i,j}^m}}}{d}_{i,j}^m{x}_{(i,j)}^{m}q+{\displaystyle \sum _{m\in M}{\displaystyle \sum _{i\in N}{\displaystyle \sum _{j\in N}{c}_j^{m,n}{\mathrm{y}}_j^{m,n}}}}q$$

(1)

$$C_2={\displaystyle \sum _{m\in M}{\displaystyle \sum _{n\in M}{\displaystyle \sum _{j\in N}\left[{t^{\prime }}_{j(m,n)}-(t_{j(m)}^{\prime }+t_j^{m,n})\right]}}}{c}_{w}q{\mathrm{y}}_j^{m,n}$$

(2)

$$E={\displaystyle \sum _{i\in N}{\displaystyle \sum _{j\in N}{\displaystyle \sum _{k\in M}q{e}_{ij}^m}}}{x}_{(i,j)}^m+{\displaystyle \sum _{i\in N}{\displaystyle \sum _{k\in M}{\displaystyle \sum _{l\in M}q{e}_i^{m,n}}}}{y}_i^{m,n}$$

(3)

Equation (1) is the cost of transportation and the cost of transit between various forms of transportation; Equation (2) is the cost of waiting at the transit node; and Equation (3) is the overall quantity of greenhouse gases released from the conversion of each road section and node.

$$\begin{array}{l}\min C^{\prime }={\displaystyle \sum _{s=1}^S{p}_s}{C}_s\\ ={\displaystyle \sum _{s=1}^S{p}_s}\left\{\begin{array}{l}{\displaystyle \sum _{m\in M}{\displaystyle \sum _{i=s}^N{\displaystyle \sum _{j=s}^N{c}_{i,j}^m}}}{d}_{i,j}^m{x}_{(i,j)}^m{q}^s+{\displaystyle \sum _{m\in M}{\displaystyle \sum _{i\in N}{\displaystyle \sum _{j\in N}{c}_j^{m,n}{\mathrm{y}}_j^{m,n}}}}{q}^s+\\ {\displaystyle \sum _{m\in M}{\displaystyle \sum _{n\in M}{\displaystyle \sum _{j\in N}\left[{t^{\prime }}_{j(m,n)}-(t_{j(m)}^{\prime }+t_j^{m,n})\right]}}}{c}_w{q}^{\mathrm{s}}{\mathrm{y}}_j^{m,n}\end{array}\right\}\end{array}$$

(4)

Equation (4) is the robust optimization’s objective function of multimodal transportation under uncertain demand.

The constraints used to describe the path optimization for multimodal transportation are as follows:

$$C_s\le (1+\gamma )C_{s*}$$

(5)

$${\displaystyle \sum _{s=1}^S{p}_s}=1$$

(6)

$${\displaystyle \sum _{m\in M}{\displaystyle \sum _{j=s,j\ne i}^N{x}_{(i,j,t_i)}^m}}-{\displaystyle \sum _{m\in M}{\displaystyle \sum _{j=d,j\ne i}^N{x}_{(i,j,t_i)}^m}}=\left\{\begin{array}{l}1,\forall i=O\\ 0,\forall i\in N\backslash \left\{O,D\right\}\\ -1,\forall i\in D\end{array}\right.$$

(7)

$${\displaystyle \sum _{m,n,l\in M}{x}_{(i,j,t_i)}^m}=1,\forall i,j\in N$$

(8)

$$x_{(i,j,t_i)}^m+x_{(j,k,t_j)}^n\ge 2y_j^{m.n}$$

(9)

$${\displaystyle \sum _{m\in M}{\displaystyle \sum _{n\in M}{y}_j^{m.n}\le 1,\forall j\in N}}$$

(10)

$$x_{(i,j,t_i)}^m\in \left\{0,1\right\},\forall m\in M;\forall i,j\in N$$

(11)

$${\mathrm{y}}_j^m\in \left\{0,1\right\},\forall m\in M;\forall i\in N$$

(12)

According to the principle of flow conservation, Equation (5) is a robust constraint, which indicates what the achievable solution’s goal for the optimal objective function under any scenario should be less than. Equation (6) indicates that the sum of the probabilities of all scenarios occurring is 1. In Equation (7), the net flow rate at the start point is equal to 1, the net flow rate at the endpoint is equal to −1 and the flow rates at all other nodes are constant. Constraint function (8) indicates that between two neighboring nodes, only one mode of transportation is available, constraint function (9) contains the mode of transportation continuity constraints, i.e., when transforming through to Output, and constraint function (10) indicates that there can be only one transit at the node. Decision variables (11) and (12) take integer variables, which can only be 0 or 1.

3.3. Model Transformation Under Different Low-Carbon Policies

Based on various low-carbon regulations, a low-carbon multimodal transportation route optimization model is constructed, i.e., using carbon trading, carbon taxes, carbon footprint requirements and carbon compensation policies as conversion tools. The total amount of carbon emissions produced by multimodal transportation are transformed into constraints, carbon emission costs or carbon emission benefits and put into the model. Correspondingly, they are labeled as Model A, Model B, Model C and Model D, which are described as follows.

I.

Model A—Choice model under mandatory carbon emission policy.

In multimodal transportation, the mandatory carbon emission policy usually includes the regulation and limitation of the carbon emissions of each transportation mode. This requires that the choice of the optimal transportation path for multimodal transportation must satisfy the constraint that the actual carbon emissions are less than or equal to the cap on carbon emissions. Therefore, under the mandatory carbon emission policy, the total carbon emissions are transformed into a constraint and modeled as follows:

$$\min C=C_1+C_2$$

(13)

$$s.t. E\le U_c$$

(14)

At the same time, Constraints (7) to (12) are established.

Equation (13) is aimed at minimizing the total cost of transportation, where the total cost of transportation is equal to the sum of the transportation cost and waiting cost. Equation (14) suggests that the overall carbon emission must be equal to or less than the carbon emission limit, where E is the total amount of carbon emissions converted by individual road segments and nodes and $U_{\mathrm{c}}$ is the upper limit of carbon emission for the transportation task under the mandatory carbon emission.

II.

Model B—Choice Model under a Carbon Tax Policy.

A carbon tax is a tax levied by the government on carbon emissions. In multimodal transportation, the government taxes each unit of actual carbon emissions at the tax rate $R_{\mathrm{t}}$. At this point, carbon emissions are converted into carbon costs and added to the total transportation costs.

$${\mathrm{minC}=\mathrm{C}}_1{+\mathrm{C}}_2{+\mathrm{R}}_{\mathrm{t}}·E$$

(15)

The constraints are the same as Equations (7)–(12).

Equation (15) indicates that the total cost of transportation is minimized under the carbon tax policy.

III.

Model C—Choice model under carbon trading policy.

A carbon trading policy means that the government sets total carbon emission allowances $U_{\mathrm{s}}$ and then allocates these allowances to enterprises or industries. Firms can buy and sell these allowances on the market; if firms emit less than their allowances, they can sell their excess allowances, while if they emit more than their allowances, they need to buy additional allowances.

$${\mathrm{minC}=\mathrm{C}}_1{+\mathrm{C}}_2{+\mathrm{R}}_s·Q_s$$

(16)

$$s.t. E+Q_s=U_s$$

(17)

$$Q_s\in R$$

(18)

At the same time, Equations (7)–(12) hold.

Goal Function (16) shows how to minimize all costs, including those associated with transportation, waiting costs and carbon trading costs or benefits. $R_s$ is the carbon trading price. Constraint (17) is the actual carbon emission and is equal to the carbon emission amount after trading. $Q_s$ is the emission quota of the transportation task under the carbon trading policy. Constraint (18) is the actual purchased carbon emission amount.

IV.

Model D—Selection Model under the Carbon Offset Policy.

The principle of carbon offsetting is similar to carbon trading, except that when the actual carbon emissions are greater than the carbon offsetting allowances $U_0$, the enterprise must purchase the difference in carbon emissions to meet the requirements and incur carbon offsetting costs. However, when the carbon emissions are less than the carbon offsetting allowances, the enterprise cannot sell the excess carbon offsetting allowances to generate revenue.

$${\mathrm{minC}=\mathrm{C}}_1{+\mathrm{C}}_2{+\mathrm{R}}_0·Q_0$$

(19)

$$s.t.{ \mathrm{Q}}_0=\left\{\begin{array}{l}E-U_0{, \mathrm{U}}_0\le E\\ 0{ ,\mathrm{U}}_0>E\end{array}\right.$$

(20)

Equation (19) denotes the total cost minimization that includes the cost of carbon offsets, where $R_0$ is the price of carbon offsets. Equation (20) denotes the non-negative constraint on the amount of carbon offsets, where $Q_0$ is the emission quota of the transportation task under the carbon offset policy.

4. Algorithm Design

The ant colony algorithm can simulate the collaborative behavior of an ant colony when solving the problem and adjust the path selection through pheromone transmission and updating, while the simulated annealing algorithm can gradually reduce the temperature during the search process, reduce the probability of accepting suboptimal solutions and help to stabilize the convergence to a more optimal solution. Combining the ant colony algorithm with the simulated annealing algorithm can better balance the global search ability and local optimization ability and improve the convergence speed and solution accuracy of the algorithm.

4.1. Simulated Annealing–Ant Colony Hybrid Algorithm

Figure 2 shows the principle of ant colony algorithm coding. As shown in Figure 2a, in a general transportation network, there is only one mode of transportation between the nodes. Whereas, in multimodal transportation networks, the node coding used by the traditional ACO algorithm is no longer applicable due to the possibility of multiple modes of transportation between neighboring nodes. Therefore, it is necessary to convert the multimodal transportation node codes to general network node codes. For the case in Figure 2b, each transportation mode of node 2 is encoded as a new node. In this way, the ACO algorithm can be used in the converted network. The converted nodes are shown in Figure 2c, where H denotes highway, R denotes railroad and A denotes aviation.

In the ant colony algorithm, due to the positive and negative feedback, the ants cannot find the globally optimal path, resulting in the algorithm easily erring towards the locally optimal solution and then the problem of “premature convergence”. However, by adding the simulated annealing mechanism into the ant colony algorithm, the algorithm can effectively prevent the problem of “premature convergence” and obtain a better convergence speed. The specific improvements are as follows: when the ants complete a cycle and generate a new solution based on the optimal route, the generation strategy is to randomly select two neighboring nodes in the generated optimal route, change the transportation mode between them and leave the rest unchanged.

4.2. Algorithmic Solution Process

The simulated annealing ant colony algorithm-specific improvement method is as follows: when the ants complete a cycle, the optimal route is obtained based on the solution to generate a new solution. The generation strategy is to generate the optimal route that is randomly selected for two neighboring nodes and change the mode of transportation between them; the rest remains unchanged. According to the idea of simulated annealing and the probability, the next step is to determine whether to accept the new route as the current optimal solution. Let the annealing temperature be $T$. The temperature changes with the number of iterations; the range of change is $[T_{\min ,}{T}_{\max }]$, and the initial temperature $T(0)=T_{\max }$. The multimodal transportation coding mode is fractional coding. In the search for a new solution to the simulated annealing operation, the generation strategy is as follows: ants carry out the optimal solution of the contemporary iteration ${\mathrm{b}}_{l-\min }(n_{l-\min ,}{c}_{l-\min })$, randomly select a route node from $n_{l-\min }$ passing route nodes and make a judgment on the fractional part of it (mode of transportation) to see whether the node has another transportation mode. If another transportation mode exists, the current transportation mode is modified from $m_1$ to $m_2$ so that the initial solution is randomly perturbed to generate a new solution ${\mathrm{b}}_{new}(n_{new,}{c}_{new})$. The objective value calculated from the new solution set is $c_{new}$, and the new solution is generated, as shown in Figure 3:

If the ant can obtain an optimal solution in one cycle, then the route it traveled is stored, denoted as $c_{\mathrm{l}-\min }$, which is the initial solution of the annealing method simulation. Then, a new solution is produced based on this solution, and the path cost of the new solution is written as $c_{new}$. Then, the change in the objective value is $\Delta c$.

$$\Delta c=c_{new}-c_{l-\min }$$

(21)

$$P=\left\{\begin{array}{l}1,\Delta c<0\\ exp(-{\displaystyle \frac{\Delta c}{T}}),\Delta c>0\end{array}\right.$$

(22)

In this case, the dynamic temperature decay coefficient [30] is introduced to hasten the convergence of the algorithm. The variation rule of the temperature decay coefficient with a given amount of iterations is as follows:

$$s\left(l\right)=0.099\left({\displaystyle \frac{\exp [0.03(-l+\frac{L}{2})]}{1+\exp [0.03(-l+\frac{L}{2})]}}+9\right)$$

(23)

where $l$ is the number of iterations and $L$ is the iteration threshold. The relationship curve is shown in Figure 4.

The particular procedure is depicted in Figure 5:

5. Calculation Analysis and Discussion

To verify the efficacy of the algorithm and model, it is presumable that the starting point is 1 and the endpoint is 25 in the multimodal transportation network shown in Figure 6. In this paper, the first 25 points of R101 in the literature [31] are selected to correspond to the coordinates of nodes in the graph, in turn, and the distances between the nodes are enlarged by 20, 10 and 10 times the distance of highway, railroad and airplane transportation distances (km), respectively. Due to economic fluctuations, seasonal changes, market trends and other factors, customer demand cannot be determined. As stated by the characteristics of the payload size distribution in reality and design in the relevant literature, it is assumed that the payload size in low-, medium- and high-payload size scenarios follows a normal distribution with a mean and variance of $q~N$ (120, 69.4), (150, 69.4) and (180, 69.4), and the likelihood of the shipment scenario amount being low, medium, or high is 0.36, 0.5 and 0.14, respectively.

The transit cost, transit time and unit loss cost between different modes of transportation are displayed in

Table 4. Transit parameters by mode of transportation.

Transportation Mode Shift	${{\mathit{t}}^{\prime }}_{\mathit{j}(\mathit{m},\mathit{n})}$/min			${\mathit{c}}_{\mathit{i}}^{\mathit{m},\mathit{n}}$/CNY			${\mathit{e}}_{\mathit{i}\mathit{j}}^{\mathit{m}}$/kg
	Road	Railroad	Airplane	Road	Railroad	Airplane	Road	Railroad	Airplane
Road	-	0.268	0.2	-	8.57	11.42	-	1.56	3.12
Railroad	0.267	-	0.35	8.57	-	17.14	1.56	-	6
Airplane	0.2	0.35	-	11.42	17.14	-	3.12	6	-

. The speed and departure time of each mode of transportation are displayed in

Table 5. Transportation speed, cost and duration by mode of transportation.

Mode of Transportation	Road	Railroad	Airplane
Velocity ($\mathrm{km}·{\mathrm{h}}^{-1}$)	90	60	600
$c_{i,j}^m$ ($\mathrm{CNY}·{\mathrm{km}}^{-1}$)	2	1	4
$e_i^{mn}$	0.2	0.004	0.19
schedule	-	8:00	9:00
		10:30	11:00
		12:00	13:00
		14:30	15:00
		17:30	17:00
		20:00	19:00

. Let it be necessary to transport the goods from node 1 to node 25 from 14:00 and deliver the goods within a specified time of 24 h or accept a penalty if it exceeds 24 h. Therefore, we must find the transportation scheme with the lowest total required transportation cost.

5.1. Algorithm Comparison

To increase the diversity problem of solutions, the simulated annealing mechanism is introduced into the improved algorithm. The pheromone volatilization factor is set as a dynamic parameter, which enhances the algorithm’s capacity for a worldwide search, due to the larger value in the beginning phases, and quickens the algorithm’s convergence, due to the smaller value in the later stage. The results show that the limitations of the traditional ACO algorithm can be solved by these two improvement steps to find more feasible solutions and search for better feasible solutions. The iterations of the two algorithms when searching for optimization are shown in Figure 7:

As can be seen in Figure 7, in the process of calculating the optimal path of the same multimodal transportation network under different policies using the ACO algorithm with identical parameter configurations, including the quantity of iterations and the number of ants, the convergence curve of the algorithm shows several drops and stagnation during the iteration process and finally falls into the best local option, while the improved algorithm can keep searching the unknown solution space due to the acquisition of more choices of different paths. The curves in the figure show that the optimization algorithm keeps searching for different solutions, resulting in an undulating average value for each round. The blue color in the figure symbolizes the mean value of the SAACO algorithm, and the orange color represents the mean value of the ACO. The average value of the SAACO algorithm is located below the ACO, which indicates that the SAACO algorithm solves well and is more stable than the ACO algorithm. In addition, the rate of convergence is quicker than the unimproved ACO algorithm.

5.2. Analysis of the Results

In this work, a simulated annealing hybrid ant colony algorithm is written in the Matlab 2021b environment to solve the example, setting the number of ants m as 10, $\alpha$ as 3, $\beta$ as 7 and the greatest quantity as 200. The designed method is applied to resolve the multimodal transportation path optimization problem considering different low-carbon policies under uncertain demand. Path optimization tests are carried out to test the low-carbon policies under each mode.

5.2.1. Low-Carbon Economy Multimodal Transportation Program Impact Analysis

Low-carbon policies often guide enterprises to reduce carbon emissions through economic instruments (e.g., $R_t$ and $Q_s$). In multimodal transport path selection, enterprises need to consider the carbon emission costs of different transportation modes to choose the most cost-effective path. Therefore, different low-carbon economies were considered to analyze the result of four low-carbon policies in multimodal transport path selection.

Table 6. Optimal scenarios under various low-carbon regulations.

Model Classification	Carbon Allowance	Carbon Tax/Carbon Price (CNY/T)	$\mathit{E}$ (kg)	$\mathit{C}$	Target Value Path (Fractional Part 1: Road; 2: Rail; 3: Air)
A	[3,4]	-	3684	86,938	1 → 7.2 → 3.1 → 4.1 → 5.2 → 10.3 → 25
B	-	400	3894	118,775	1 → 7.2 → 3.1 → 4.1 → 5.2 → 10.2 → 15.3 → 25
C	5t	70	3708	88,418	1 → 7.2 → 3.1 → 4.1 → 5.2 → 10.2 → 15.3 → 25
D	5t	70	3756	114,038	1 → 7.2 → 3.1 → 4.1 → 5.2 → 10.2 → 15.3 → 25

and

Table 7. Percentage of different modes of transportation.

Mode of Transportation	Proportion
Road	34.8%
Rail	47.8%
Airplane	17.4%

record the different target values as well as transportation options under various low-carbon regulations.

As can be seen from Table 5, A (mandatory carbon emission) has the lowest economic cost, which is because mandatory carbon emission policies usually set strict carbon emission limits. The policy of carbon taxation has the most direct and significant impact on transportation costs. This is because the carbon tax policy directly imposes economic penalties on the carbon emission behavior of enterprises and increases their operating costs. In terms of path selection, Model A minimizes the transportation path because carbon emission is a mandatory constraint. The three policies of a carbon tax, carbon trading and carbon offset policy multimodal transportation path selection impact are not significant; the results are mainly dominated by economic factors, suggesting the path with the lowest overall expense.

From the proportion of various modes of transportation in Table 6, the road transportation mode accounted for 34.8% of the proportion, railroad transportation accounted for 47.8% of the proportion and air transportation accounted for only 17.4%. That is, the multimodal transportation scheme focuses on choosing railroad for transportation, which fully shows that railroad transportation has significant advantages in reducing carbon emissions and costs, while air transportation exists only in a few nodes. However, when choices are made at the nodes, less air transportation is chosen for intermodal transportation because its costs and carbon emissions are higher than the first two modes of transportation. Therefore, although road transport has a bright spot in terms of flexibility and convenience, it lacks advantages in long-distance transport compared with railroads, which are more advantageous in dealing with the path optimization problem under different low-carbon policies and can better balance the objectives.

5.2.2. Analysis of Carbon Emissions and Economic Costs of a Low-Carbon Economy

In order to investigate the changes in transportation costs, carbon emissions and carbon emission costs due to multimodal transport path optimization when the carbon tax is changed, a change in the carbon tax is made. The results are in

Table 8. Impact of different carbon taxes on multimodal transport path optimization.

${\mathit{R}}_{\mathit{t}}$ (CNY/t)	${\mathit{C}}^{\prime }$ (CNY)	$\mathit{E}$ (kg)	Cost of Carbon Emissions	Path (Fractional Part 1: Road; 2: Railroad; 3: Airplane)
0	86,834	3669	0	1 → 7.2 → 3.1 → 4.1 → 5.2 → 10.2 → 15.3 → 25
300	99,547	3894	1168.2	1 → 7.2 → 3.1 → 4.1 → 5.2 → 10.2 → 15.3 → 25
400	118,775	3894	1557.6	1 → 7.2 → 3.1 → 4.1 → 5.2 → 10.2 → 15.3 → 25
500	124,723	3871	1935.5	1 → 7.2 → 3.1 → 4.1 → 5.2 → 10.3 → 25
1000	127,435	3756	3756	1 → 7.2 → 3.1 → 4.1 → 5.2 → 10.3 → 25

.

As shown in

Table 9. Solution results under the carbon trading policy.

Number	${\mathit{U}}_{\mathit{s}}$	${\mathit{R}}_{\mathbf{s}}$	Total Transportation Costs	Actual Carbon Emissions	Costs/Benefits of Carbon Emissions
1	0.1t	70 CNY/t	114,256	3669	249.83
2	1t	70 CNY/t	107,335	3734	191.38
3	5t	70 CNY/t	107,064	3870	−79.1
4	8t	70 CNY/t	106,849	3799	−294.07
5	10t	70 CNY/t	106,706	3760	−436.8

and

Table 10. Solution results under carbon offset policy.

Number	${\mathit{U}}_{\mathit{s}}$	${\mathit{R}}_0$	Total Transportation Costs	Actual Carbon Emissions	Cost of Carbon Emissions
1	0.1t	70 CNY/t	115,374	3842	261.94
2	1t	70 CNY/t	111,296	3899	202.93
3	5t	70 CNY/t	119,182	3756	350
4	8t	70 CNY/t	119,380	3756	560
5	10t	70 CNY/t	119,520	3756	700

, with the increase in carbon emission quota under the carbon trading policy, the total transportation cost decreases, which is due to the fact that the actual carbon emissions are smaller than the quota and the saved quota is converted into revenue. While the carbon emission cost under the carbon offset policy gradually increases when the allotted amount for carbon emissions exceeds 1 ton, the carbon offset mechanism cannot effectively encourage the decision-makers of multimodal transportation to choose low-emission solutions, and it is necessary to promote emission reduction with the help of other low-carbon policies. As can be seen in Figure 7, with the increase in the carbon quota, the total cost changes very little; this is due to the small proportion of carbon emission costs and the overall expense, as well as its impact on the total cost, which is negligible.

Based on distinctive low carbon policies, observing the changes in the optimal transport path of multimodal transport operators and summarizing its law, we can find that the mandatory carbon emission policy requires that carbon emissions must be less than or equal to the carbon emission limit, which is a constraint. In addition, operators must comply with the conditions when they formulate transport paths, which is a mandatory means in terms of emission reduction. Under the carbon trading and carbon compensation policy, the actual carbon emissions exceeding part of the quota generate the cost of carbon emissions. The carbon tax taxes all the actual carbon emissions. Carbon trading, carbon compensation and carbon tax policies all belong to the scope of the objective function, and the size of the carbon emission costs in the economic cost of the proportion of the optimal transportation path selection has a greater impact. Therefore, the impact of reducing emissions and saving energy mandatory carbon emissions is the optimal solution of the four different low-carbon policies.

5.3. Case Study

To validate the effectiveness of the models developed and algorithms designed in this paper, taking a public rail–fly multimodal transportation system in the eastern region of China as the background, 17 cities are selected as the nodes of the transportation network. To facilitate the representation of the transportation scheme below, the actual network nodes are numbered, as shown in Table 10, from 1 to 17. The highway distance data in the figure are derived from the Gaode map; the railroad distance from the China Railway Network and the air transportation distance comes from the flight mileage of China Southern Airlines. The relevant data are shown in

Table 11. Multimodal transportation city numbers.

Serial Number	City	Serial Number	City	Serial Number	City
1	Guangzhou	7	Hefei	13	Jinan
2	Fuzhou	8	Shanghai	14	Taiyuan
3	Changsha	9	Nanjing	15	Shijianzhuang
4	Nancang	10	Xuzhou	16	Tianjin
5	Hangzhou	11	Zhengzhou	17	Beijing
6	Wuhan	12	Handan

. Figure 8 shows the railroad transportation routes of the transportation network, and nodes 0 and 17 are the starting and ending points of the transportation tasks. Due to the excessive number of road and air transportation routes, they are not shown in the figure. A route selection scheme with the lowest total cost of transportation under the low-carbon policy is sought.

As can be seen from Figure 9, among the four kinds of carbon emission policies, the mandatory carbon emission policy means are tough. The effect of energy saving and emission reduction is the most significant, and the cost is the lowest. The three kinds of policy means, namely, carbon tax, carbon trading and carbon compensation, are more moderate, and the effect of energy saving and emission reduction is the second-most effective. The optimal path scheme under mandatory carbon emissions is shown in Figure 10.

6. Conclusions

Based on the perspective of a low-carbon economy, this paper constructs a robust optimization model for multimodal transport reflecting different low-carbon policies under demand uncertainty. The characteristics of different low carbon policies and uncertain demand of cargo transportation are taken into account, as well as the fact that railroads have departure time restrictions and airplanes have no-fly periods. The simulated annealing–ant colony hybrid algorithm is designed to solve the problem, and the established multimodal transportation robust path optimization model is verified by example. The comparison of the algorithms shows that the improved algorithm is feasible. Meanwhile, the following conclusions are drawn:

(1)

In the process of multimodal transportation, the transportation cost and carbon emissions under different low-carbon policies will also change. With different departure times, the transportation cost will also change accordingly. This study provides new ideas for multimodal transportation path selection under different low-carbon policies, while the designed algorithm is applicable to multimodal transportation path optimization problems under uncertain demand, which can produce better solutions with better robustness. It can also be extended and applied to other shortest-path problems.

(2)

In future research, the dual uncertainty in transportation time and transportation demand can be considered, and the carbon emission problem can be combined with the nature of goods, such as agricultural products, dangerous goods, etc. Carbon tax (carbon price), time constraints, economic costs and other factors can be taken into account so that optimal decision-making can be carried out and the environment can be protected at the same time.