Path Optimization of Container Multimodal Transportation Considering Differences in Cargo Time Sensitivity

Abstract

To optimize the container multimodal transportation path selection problem with differences in cargo time sensitivity, this paper introduces the concept of mixed time windows and establishes an integer programming model from the perspective of carriers. The objective is to minimize the total transportation cost while meeting the customer’s on-time delivery requirements. The model imposes hard time window constraints on cargoes with on-time delivery requirements and soft time window constraints on general cargoes. A bi-level genetic algorithm is employed to solve the model. A case study is conducted to demonstrate the effectiveness of the proposed approach. Experimental results show that the bi-level genetic algorithm’s convergence ability, efficiency, stability, and global search capabilities are far superior to those of traditional single-level genetic algorithms. This indicates that the use of multi-stage solution approaches can mitigate the shortcomings of genetic algorithms in dealing with high-dimensional problems. In addition, when analyzing the effects of time penalty intensity and time window on decision-making results, it is found that the more time-sensitive cargoes are more likely to be affected by the time window setting, and will tend to sacrifice costs to ensure that the time demand is met. At the same time an increase in demand for time-sensitive services is accompanied by an increase in operating costs. Accurate assessment and classification of time sensitivity of cargoes can effectively prevent wasting resources in unnecessary areas, and ultimately maximize decision outcomes and ensure the most efficient use of resources.

Keywords

freight systems intermodal freight transport optimization transportation network modeling algorithms

Under the impetus of economic globalization, globalization of production and consumption is becoming increasingly common, and consumer demand for long-distance transportation is growing rapidly, leading to a continuous rise in logistics and transportation costs ( 1 ). Therefore, multimodal transportation that can integrate the advantages of different transportation resources, provide supplier-to-customer chain services, and reduce environmental and energy consumption has gradually become one of the logistics strategies vigorously promoted by various countries. Container transportation occupies an important position in multimodal transportation, as it can significantly improve transportation efficiency and reduce transportation costs ( 2 ).

Scholars, both in China and other countries, have conducted extensive research on optimizing multimodal transportation paths for containers. To adapt to the ever-increasing demand for efficient production and operation in modern society, time constraints have received increasing attention from researchers. As a result, scholars have begun to include both hard ( 3 , 4 ) and soft time window ( 5 , 6 ) constraints in their solution models. However, considering the complexity and diversity of transportation environments in practical situations, a single time window gradually becomes inadequate to meet the practical requirements of path optimization problems. Therefore, mixed time windows have been studied. When designing mixed time window models for multimodal transportation, scholars from both the Chinese research field and from other countries have taken various approaches. Ding ( 7 ) established a mixed time window by setting flexible highway transportation without time window constraints, relatively fixed railway and waterway transportation constrained by hard time windows, and consignees with relatively free receiving time constrained by soft time windows, based on different delivery time requirements for transportation modes. Qu et al. ( 8 ) proposed a new time penalty function that takes into account the characteristics of both hard and soft time windows. This function imposes time penalty costs on materials that cannot be delayed and must arrive before the soft time window constraints. The function imposes a hard time window constraint on late-arriving materials, preventing them from being delayed, while applying a soft time window constraint on early-arriving materials, thereby compelling them to incur additional penalty costs. Meanwhile, Ongcunaruk et al. ( 9 ) imposed mixed time window constraints by setting different time windows for different service areas based on the actual situation of the destination node.

At the same time, with increasing customer demand for cargoes and improved service levels, customized and personalized services are becoming more and more popular among consumers. Hu et al. ( 10 ) demonstrated a positive correlation between customized logistics services and customer satisfaction. Therefore, differentiated services have become an important trend for the development of the logistics industry. Moreover, consumers now require logistics services to be punctual rather than simply fast. Liu et al. ( 11 ) also pointed out that customers’ sensitivity to waiting is a key factor in service providers’ decision making and customers’ queue-joining behavior. Therefore, considering time sensitivity ( 12 ) and incorporating on-time delivery requirements into the problem of multimodal route optimization is important to provide decision support to carriers.

Since the path optimization problem is an Non-deterministic polynomial-time hard problem, it is usually solved by metaheuristic algorithms. Genetic algorithms, as a classical branch of metaheuristic algorithms, have been widely used in several related fields ( 13 ). Compared with other optimization methods, the advantage of genetic algorithms lies in their unique parallelism and probabilistic nature, which is convenient for global optimization ( 14 ). However, with the increasing scale and complexity of research problems in modern society, traditional heuristic algorithms have gradually failed to meet the needs of solving large-scale, multivariate, and multi constrained complex problems. The complexity of genetic algorithms increases when the number of iterations increases, and this computational drawback is particularly obvious when dealing with complex problems with large-scale solution spaces ( 15 ). Therefore, the study of improved and hybrid algorithms has become a major research topic for scientists. Genetic programming, as an extension of genetic algorithms, is also a direction of algorithm improvement ( 16 ). Bi-level genetic algorithm programming involves two levels of hierarchy: the upper-level problem is associated with a decision maker with a higher hierarchy and the nested problem is associated with a decision maker at a lower level ( 17 ). Liu et al. ( 18 ) designed a bi-level genetic algorithm that divides the problem into two phases by planning the sub-partition routes and then the routes connecting the various sub-partitions, nesting the large problem into two sub-problems. The algorithm effectively solves the transportation problem in different regions within the manufacturing plant, and the overall transportation time is slightly better than the existing results. The problem dealt with in this paper can also be partitioned into a two-layer problem. In the context of adding on-time delivery service to multimodal carriers, customers with this value-added service will inevitably need to have a higher service priority. Therefore, setting the on-time delivery problem as an upper-level problem with higher priority and setting the general cargo as a nested problem with lower level is consistent with the practical situation. Accordingly, this paper designs a bi-level genetic algorithm that meets the characteristics of the research problem in this paper.

In summary, there is currently limited research on the multimodal transportation path optimization problem with mixed time window constraints, based on differences in cargo time sensitivity. Differentiated services are a major trend in modern logistics development. Therefore, this paper attempts to apply different time window constraints to on-time delivery and general delivery respectively, based on the varying sensitivity of different cargoes to time constraints. A mixed time window penalty function is designed for this purpose, and an integer programming model is constructed with cost minimization as the objective. A bi-level genetic algorithm is then proposed to solve the multimodal transportation path optimization problem, while meeting the differentiated time requirements of on-time delivery and general cargoes simultaneously.

Problem Description

Model Description

Assuming that a multimodal transport carrier has a batch of containerized cargoes that need to be transported to their destination, including a portion of cargoes with on-time delivery demands, the required quantities of on-time delivery cargoes and general cargoes have already been determined. The cargo transportation process needs to pass through multiple nodes, and various transportation modes can be adopted between adjacent nodes. As shown in Figure 1, O and D represent the starting point and the end point, respectively. This paper assumes that only highway, railway, and waterway modes of transportation can be used to connect intermediate nodes, and the time and cost required for different transportation and transfer modes are different. Additionally, with the limited transportation resources of some cities, only certain transportation modes are available for use in certain nodes, and there are upper limits on the transportation capacity and transfer capability of different transportation modes between nodes.

Figure 1.

Map of multimodal transport network concept.

In previous literature, scholars rarely used differences in the time sensitivity of cargoes as a starting point to construct a mixed time window. They tended to focus on constructing mixed time window functions based on different transportation modes ( 7 ), or based on the distinction between early and late penalties ( 8 ), or the differences between nodes ( 9 ). However, customized and diversified services have gradually become an important way to improve market competitiveness, and customer integration is crucial for improving supply chain performance ( 19 ). Therefore, we chose to classify cargoes into time-sensitive and general cargoes based on their time sensitivity, and imposed hard time window constraints on time-sensitive cargoes and soft time window constraints on general cargoes, to construct a mixed time window. Specifically, the hypothesis is that the types of cargoes are denoted by M = {A, B}, where A represents time-sensitive cargoes with a punctual delivery requirement and B represents general cargoes without a punctual delivery requirement. To consider the two types of cargoes simultaneously in the computation, this paper refers to the penalty cost design of Zhou and Zhao ( 20 ), which gradually increases as the arrival time exceeds the optimal service time, and takes infinity after exceeding the earliest or latest time allowance. The mixed time window penalty function, is derived from the deformation of the mixed time window model with infinite penalty cost taken after exceeding the earliest or latest time allowance, and differentiated time constraints are applied to time-sensitive and general cargoes (Figure 2). Assuming that the arrival time of cargo at the destination is denoted as $t_{j}, (j = D)$ , the hard time window constraint is $[{ET}_{A}, {LT}_{A}]$ , and the maximum constraint for the soft time window is $[{ET}_{B}, {LT}_{B}]$ , with the optimal service time being $[{et}_{B}, {lt}_{B}]$ . When cargo has an on-time delivery requirement, the hard time window constraint is adopted. If $t_{j} < {ET}_{A}$ or $t_{j} > {LT}_{A}$ , a penalty function is set to an infinite value. When cargo does not have an on-time delivery requirement, the soft time window constraint is adopted. Cargoes that do not arrive within the optimal service time $[{et}_{B}, {lt}_{B}]$ will incur penalty costs because of the maximum holding cost $C_{e}$ for early arrival and the maximum penalty cost $C_{l}$ for delayed arrival.

Figure 2.

Penalty function of mixed time windows: (a) hard time window for cargo arriving on time and (b) soft time window for general cargo.

Therefore, the purpose of this paper is to consider the optimization objective of minimizing the total transportation cost, introduce mixed time window constraints, and ultimately obtain the optimal path selection for container multimodal transportation under the mixed time constraints of on-time delivery and non-on-time delivery requirements.

Model Hypothesis

It is assumed that:

No accidents occur at any point during the transportation process, and that there is no need to consider cargo damage.

The transportation modes available for use include waterway, highway, and railway transport.

Cargo will be transported using only one mode of transportation between two transport nodes and cannot be divided.

Transportation mode changes are only permitted at designated node cities on the transportation network.

At each transfer node, the cargo may undergo at most one change in transportation mode.

It is assumed that the order has been placed and that the types and quantities of cargo to be transported to each destination are known.

Cargo is shipped in a standard 20-foot container.

Modeling

Model Parameters

The sets, parameters, and decision variables are shown in Tables 1 to 3.

Table 1.

Set and Indices Description

Set and indices	Description
N	Set of transportation nodes ( $N = {1, 2, 3 \dots n}$ , where 1 represents the origin O and n represents the destination D)
K	Set of transportation modes ( $K = {1, 2, 3}$ )
M	Set of cargo types ( $M = {A, B}$ )
F	Set of transportation arcs. When $(i, j, k) \in F$ , it indicates a connection from i to j via transportation mode k. Conversely, if $(i, j, k) \notin F$ , it indicates that there is no connection from i to j via transportation mode k.
$i, j, h$	Transportation nodes
$k, l$	Transportation modes. k or l = 1,2,3 represent the modes of transportation by highway, railway, and waterway respectively.
$m$	Cargo types. m = A represents cargoes with on-time delivery requirement, m = B represents general cargoes.

Table 2.

Parameter Description

Parameter	Description
${CP}_{ij}^{k}$	The unit cost of transportation from node i to node j via transportation mode k
${CS}_{i}^{kl}$	The unit cost of transshipment at node i from transport mode k to transport mode l
$C T_{m}$	The unit time penalty cost for cargo of type m
$q_{m}$	The shipment volume of cargo of type m
${QP}_{ij}^{k}$	The maximum transportation capacity between nodes i and j via transportation mode k
${QS}_{i}^{kl}$	The maximum transshipment capacity of node i in transit between transport mode k and the other l
$d_{ij}^{k}$	The transportation distance of cargoes from node i to node j via transportation mode k
$v_{k}$	The transportation speed of transportation mode k
${tp}_{ij}^{k}$	The transportation time of cargoes from node i to node j via transportation mode k
${ts}_{i}^{kl}$	The time required for transshipment of cargoes at node i from transportation mode k to transportation mode l
$t t_{m}$	The total transportation time for cargo of type m
$C_{e}$	The maximum holding cost per unit for early arrival of cargoes
$C_{l}$	The maximum penalty cost per unit for late arrival of cargoes
$E T_{m}$	The earliest permissible arrival time for cargoes of type m at the destination
$L T_{m}$	The latest permissible arrival time for cargoes of type m at the destination
$[e t_{B}, l t_{B}]$	The optimal service time for cargoes of type B

Table 3.

Decision Variable Description

Decision variable	Description
$x_{mij}^{k}$	Binary, the value is 1 if cargoes of type m are shipped from node i to node j via transportation mode k, else the value is 0
$y_{mi}^{kl}$	Binary, the value is 1 if cargoes of type m are transferred from transport mode k to transportation mode l at node i, else the value is 0

Integer Programming Model

The optimization process of container multimodal transport paths can be interpreted as ensuring the minimization of transportation total costs for cargo from the origin to destination port, while optimizing the operational efficiency during transportation to reduce time penalty costs. This model can be defined as:

\begin{matrix} minZ = & \sum_{m \in M} \sum_{i, j \in N} \sum_{k \in K} {CP}_{ij}^{k} q_{m} x_{mij}^{k} + \sum_{m \in M} \sum_{i, j \in N} \sum_{k, l \in K} {CS}_{i}^{kl} q_{m} y_{mi}^{kl} \\ + \sum_{m \in M} C T_{m} q_{m} \end{matrix}

(1)

subject to

\sum_{i, j \in N} \sum_{k \in K} x_{mij}^{k} = 0, \forall (i, j, k) \notin F, \forall m \in M

(2)

\sum_{j \in N} \sum_{k \in K} x_{mij}^{k} \leq 1, \forall i = {2, \dots, n - 1}, \forall m \in M

(3)

\sum_{j \in N} \sum_{l \in K} x_{mij}^{k} = 1, i = 1, \forall m \in M

(4)

\sum_{i \in N} \sum_{l \in K} x_{mij}^{k} = 1, j = n, \forall m \in M

(5)

\sum_{h \in N} \sum_{k \in K} x_{mhi}^{k} - \sum_{j \in N} \sum_{l \in K} x_{mij}^{l} = 0, \forall i = {2, \dots, n - 1}, \forall m \in M

(6)

\sum_{k, l \in K} y_{mi}^{kl} \leq 1, \forall i = {2, \dots, n - 1}, \forall m \in M

(7)

\sum_{h \in N} x_{mhi}^{k} = \sum_{l \in K} y_{mi}^{kl}, \forall i = {2, \dots, n - 1}, \forall k \in K, \forall m \in M

(8)

\sum_{j \in N} x_{mij}^{l} = \sum_{k \in K} y_{mi}^{kl}, \forall i = {2, \dots, n - 1}, \forall l \in K, \forall m \in M

(9)

\sum_{m \in M} q_{m} x_{mij}^{k} \leq {QP}_{ij}^{k}, \forall i, j \in N, \forall k \in K

(10)

\sum_{m \in M} q_{m} y_{mi}^{kl} \leq {QS}_{i}^{kl}, \forall i = {2, \dots, n - 1}, \forall k, l \in K

(11)

{tp}_{ij}^{k} = \frac{d_{ij}^{k}}{v_{k}}, \forall i, j \in N, \forall k \in K

(12)

t t_{m} = \sum_{i, j \in N} \sum_{k \in K} {tp}_{ij}^{k} x_{mij}^{k} + \sum_{i \in N} \sum_{k, l \in K} {ts}_{i}^{kl} y_{mi}^{kl}, \forall m \in M

(13)

C T_{m} = {\begin{matrix} \infty & , & {tt}_{m} < {ET}_{A} \\ 0 & , & {ET}_{A} \leq {tt}_{m} < {LT}_{A} \\ \infty & , & {LT}_{A} \leq {tt}_{m} \end{matrix} \begin{matrix}  \end{matrix}, m = A

(14)

C T_{m} = {\begin{matrix} \begin{matrix} C_{e} & , & {tt}_{m} < E T_{B} \\ C_{e} \cdot \frac{e t_{B} - {tt}_{m}}{e t_{B} - E T_{B}} & , & E T_{B} \leq {tt}_{m} < e t_{B} \\ 0 & , & t_{B} \leq {tt}_{m} < l t_{B} \end{matrix} \\ \begin{matrix} C_{l} \cdot \frac{{tt}_{m} - l t_{B}}{L T_{B} - l t_{B}} & , & l t_{B} \leq {tt}_{m} < L T_{B} \\ C_{l} & , & L T_{B} \leq {tt}_{m} \end{matrix} \end{matrix} \begin{matrix}  \end{matrix}, m = B

(15)

E T_{j} \leq e t_{j} < l t_{j} \leq L T_{j}, j = n

(16)

x_{mij}^{k} = {0, 1}, y_{mj}^{kl} = {0, 1}, \forall i, j \in N, \forall k, l \in K, \forall m \in M

(17)

Equation 1 represents the objective function, which seeks to minimize the transportation costs, transfer costs, and time penalty costs incurred along the route. In the block of constraints, Equation 2 excludes impassable paths. Equation 3 indicates that at most one transportation mode can be selected between adjacent nodes. Equations 4 to 6 represent the node flow balance and define the relationship between the starting point and the ending point. Equation 7 denotes that the same batch of cargoes can only be transited once at one node. Equations 8 and 9 indicate that if a changeover occurs at node i, the transportation arc adjacent to node i must be selected, and the converse holds. Equations 10 and 11 are capacity constraints, where Equation 10 represents that the total amount of cargoes transported on the path will not exceed the capacity limit of the transportation arc on the path, and Equation 11 represents that the total amount of cargoes on the path will not exceed the transfer capacity limit of the transportation node on the path; Equation 12 is used to calculate the transportation time between two adjacent nodes; Equation 13 represents the arrival time of the cargoes at the destination; Equations 14 and 15 represent the mixed time window penalty function; Equation 16 represents the timing relationship under the time window constraint; Equation 17 indicates that the decision variable is a 0–1 variable.

Bi-Level Genetic Algorithm

Genetic Manipulation Description

The path optimization problem studied in this paper falls within the realm of network optimization. Network optimization is a combinatorial optimization problem and is a typical Non-deterministic polynomial-time hard problem. Precise algorithms cannot resolve the contradiction between the time complexity and the quality of the solution for such problems. It is challenging to determine the global optimum within an acceptable time frame. The genetic algorithm, as a type of random search algorithm, performs well in handling such problems ( 21 ). On the other hand, because of the different demands and priorities of on-time delivery cargoes and general cargoes, by decomposing the problem into two sub-problems for on-time delivery cargoes and general cargoes, we can not only optimize precisely for different types of cargoes but also effectively reduce the complexity of the problem, making it easier to understand and solve. This dual-layer structure design makes the algorithm more in line with the characteristics of the actual problem, and also improves the accuracy and feasibility of the solution. Therefore, this paper adopts a bi-level genetic algorithm to solve the model.

Since the time penalty function for the inability to complete the on-time demand is set to infinity when studying the container multimodal route optimization problem considering the time-sensitive differences in cargo, achieving on-time delivery becomes a top priority. Therefore, in this paper, we set the multimodal transportation path optimization problem of on-time cargo as the solution objective of the upper-level genetic algorithm, and the multimodal transportation path optimization of general cargo as the solution objective of lower-level genetic algorithm in designing the bi-level genetic algorithm. Ultimately, the optimal path and transportation mode selection scheme is integrated with the premise of satisfying the on-time delivery requirements. The specific algorithmic steps are as follows:

Chromosome encoding and decoding: The multimodal transportation path optimization problem is a typical Non-deterministic polynomial-time hard problem. Traditional binary encoding methods often result in many infeasible solutions, thereby significantly reducing the convergence speed. Therefore, this paper adopts a real number encoding method, dividing the chromosome into two segments: the first segment encodes the transportation path and the second segment encodes the transportation mode. Figure 3 shows the chromosome encoding methods for both the upper- and lower-level genetic algorithms, while Figure 4 shows the resulting multimodal transportation path. This method increases the convergence speed and efficiency.

Population initialization: Firstly, we set the starting point as node 1 and the endpoint as node n. Then, we generate a part of the transportation path encoding using a random sampling method based on the directed network graph. Each node randomly connects to an accessible adjacent node until it connects to node n. Subsequently, based on the initial path of length x generated in the previous step, we produce a combination of transportation modes of length x–1. Here, the first mode of transportation is randomly selected from the available modes between the first and second nodes, and so forth. Finally, we eliminate infeasible solutions based on the transportation and transshipment upper limit constraints of the model. This process of generating initial feasible solutions is repeated until the population size P is reached.

Chromosome fitness computation: The upper-level genetic algorithm’s objective function is influenced by the time window penalty function, leading to solutions with infinite values. To address this, objective function values are ranked, and the fitness determined using rank-based assignment. Lower objective function values receive higher ranks and higher fitness values.

Selection: The roulette wheel method is used to increase the probability of selecting individuals with higher fitness as parents.

Crossover: Using the single-point crossover method ( 22 ), random intersections are sought between transportation nodes and transportation modes for crossover processing (Figure 5). In the case of infeasible solutions generated, repairs are conducted as follows: when the intersection occurs within the transportation route segment. If a node cannot be connected to its neighboring node directly but can be connected through other nodes, the Floyd algorithm is utilized to establish connectivity between the nodes. If a node cannot be connected to its neighboring node by any means, attempts are made to connect it to a more distant node using the Floyd algorithm until there is a connection between the nodes. When the crossover occurs within the chromosome segment that determines the transportation mode. If a transportation mode fails to bridge the gap between two adjacent nodes, another feasible transportation mode is randomly selected. In addition to this, capacity constraints need to be detected on the newly generated children. If the constraints cannot be satisfied then the segments are re-selected until a feasible solution is generated or the conditions for dropping the crossover are met.

Mutation: The replacement mutation method was used and some genes were randomly selected for mutation based on the mutation probability $γ$ (Figure 6). Then, the generated offspring are checked for feasibility. If the new offspring is an infeasible solution, it is repaired or re-mutated using the repair method mentioned in the crossover until a feasible solution is obtained.

Elitism strategy: After completing the crossover and mutation operations, the elitism strategy is used to select superior individuals from the combined parent and offspring populations to enter the next generation ( 23 ). This ensures that the good individuals in the parent population are not lost during the evolutionary process.

Figure 3.

Chromosome coding schematic diagram.

Figure 4.

Chromosome decoding schematic diagram.

Figure 5.

Chromosome crossover schematic diagram.

Figure 6.

Chromosome mutation schematic diagram.

Algorithmic Processes

The flowchart of the bi-level genetic algorithm is shown in Figure 7.

Figure 7.

Bi-level genetic algorithm flowchart.

Computational Experiments and Results

Case Description

This section selects part of the data from the web portal of Jincheng Logistics as an example, setting up a multimodal transportation network consisting of 10 nodes to analyze the effectiveness of the model and algorithms. A batch of cargoes with predefined volume will be transported from origin O to destination D, including some with strict time constraints represented by a hard time window constraint of [40,70] (in hours) and others with soft time window constraints of [30,40,70,80]. The maximum penalty cost for delayed delivery of a unit of general cargo and the maximum storage cost for early delivery of a unit are set at 50,000 yuan/Twenty-foot Equivalent Unit (TEU) and 30,000 yuan/TEU, respectively. There are up to three transportation modes available between any two nodes, namely, highway, railway, and waterway, with transportation speeds of 76, 60, and 20 km/h, respectively. The connection relationships between the nodes are shown in Figure 8. Tables 4 and 5 represent the transportation distance and maximum transport capacity of different transportation modes between different nodes, respectively. Table 6 represents the maximum transfer capacity at each node, while Table 7 shows the transit time required for different transportation modes.

Figure 8.

Diagram of example multimodal transport network.

Table 4.

Transport Distance Between Nodes (kilometers)

Node pair	Highway	Railway	Waterway
O→2	394	303	376
O→3	352	429	661
O→4	432	628	557
2→5	1,016	1,109	1,185
2→6	1,107	1,173	846
3→5	874	1,194	NA
3→6	894	972	NA
4→5	822	795	1,000
4→6	554	577	665
5→7	242	515	237
5→8	717	NA	670
5→9	580	456	554
6→7	481	996	574
6→8	960	NA	959
6→9	823	581	837
7→D	439	535	524
8→D	124	NA	198
9→D	136	147	159

Note: O = origin; D = destination; NA = not available.

Table 5.

Upper Limit of Internode Traffic (Twenty-foot Equivalent Unit)

Node pair	Highway	Railway	Waterway
O→2	160	180	200
O→3	190	200	175
O→4	168	190	185
2→5	110	200	210
2→6	210	220	190
3→5	180	195	0
3→6	100	110	0
4→5	220	230	210
4→6	170	140	200
5→7	150	195	182
5→8	90	0	100
5→9	180	200	210
6→7	230	220	230
6→8	400	0	390
6→9	220	190	198
7→D	180	220	215
8→D	215	228	0
9→D	200	220	219

Note: O = origin; D = destination.

Table 6.

Upper Limit of Node Transfer Capability (Twenty-foot Equivalent Unit)

Node	Highway–Railway	Railway–Waterway	Highway–Waterway
O	NA
2	200.00	210.00	220.00
3	170.00	210.00	230.00
4	160.00	190.00	180.00
5	180.00	165.00	210.00
6	210.00	220.00	180.00
7	190.00	180.00	210.00
8	0.00	190.00	0.00
9	200.00	190.00	210.00
D	NA	NA	NA

Note: O = origin; D = destination; NA = not available.

Table 7.

Transit Time Between Modes of Transport (hours)

Transport mode	Highway	Railway	Waterway
Highway	0	1	1
Railway	1	0	2
Waterway	1	2	0

The equations for calculation of in-transit transportation unit cost for highways and railways, according to the International Rules for the Collection of Transport Charges for Containerized Motor Vehicles and Rules for the Transportation of Containerized Cargoes by Railways, are shown in Equation 18.

{CP}_{ij}^{k} = p_{2}^{k} \cdot d_{ij} + p_{1}^{k}, \forall k = 1, 2

(18)

where $p_{1}^{k}$ represents the transport base price 1 for transportation mode k, and $p_{2}^{k}$ represents transport base price 2 for transportation mode k.

For waterway transportation, a box rate of 950 yuan/TEU will be applied. As a result, the unit freight rate for containerized cargo by water transportation can be derived as shown in Equation 19.

{CP}_{ij}^{k} = 950, \forall k = 3

(19)

The transportation prices for each mode of transportation are presented in Table 8.

Table 8.

Price of Transportation by Mode of Transportation

Transport mode	Highway	Railway	Waterway
Transport base price 1 (yuan/TEU)	15	440	NA
Transport base price 2 (yuan/TEU·km)	8	3.185	NA
Freight charge (yuan/TEU)	NA	NA	950

Note: TEU = Twenty-Foot Equivalent Unit; NA = not available.

Moreover, since the transfer costs generated during freight transfer have a relatively low correlation with the transportation mode, and the emphasis is on the expenditure of loading and unloading costs, the transfer costs in this paper will be based on the rate table for loading, unloading, and handling operations of railway vehicle cargo at 195 yuan/TEU, and the transfer costs will not be calculated if there is no change in transportation mode.

Analysis of Algorithm Effectiveness

To verify the effectiveness of the bi-level genetic algorithm designed in this paper to solve the multimodal transportation path selection problem considering the difference in cargo time sensitivity, this study evaluates the effectiveness of the bi-level genetic algorithm. This is done by comparing the bi-level genetic algorithm and the single-level genetic algorithm, respectively, from the four aspects of the algorithm’s basic parameter sensitivity, the solution time, the convergence speed, and the stability of the solution quality. Considering that the genetic algorithm is a heuristic algorithm, the convergence result has a certain degree of randomness. Therefore, the algorithm is looped 10 times each time, and the optimal solution and its performance evaluation indexes are selected for comparison.

First, we assess the sensitivity of both algorithms to the fundamental parameters of the algorithm. In this study, we set the crossover probability ( $λ$ ) to 0.9 and the mutation probability ( $γ$ ) to 0.05. We then compared the optimal results achieved by both the bi-level genetic algorithm and the single-level genetic algorithm under three distinct scenarios. These scenarios comprised: (i) a population size (P) of 30 and an iteration parameter (G) set to 100, (ii) P of 60 with G of 200, and (iii) P of 100 with G of 300. The optimal solutions derived from the algorithm under these various conditions, as well as their iterative results, are depicted in Figure 9 and Table 9. It is evident that the sensitivity of the bi-level genetic algorithm to the basic parameters is significantly lower compared with the single-level genetic algorithm. This disparity arises from the bi-level genetic algorithm’s enhanced capability to harmonize global and local search trade-offs through the synergistic impact of its upper and lower levels, thus contributing to a more consistent algorithmic performance. Consequently, the bi-level genetic algorithm demonstrates superior solution outcomes when dealing with small population sizes and a limited number of iterations.

Figure 9.

Diagram of algorithm convergence iteration under different scenarios of population size (P) and iteration parameter (G): (a) P = 30, G = 100, (b) P = 60, G = 200, (c) P = 100, G = 300.

Table 9.

Optimal Solution and Computation Time Under Different Scenarios

Scenario	Bi-level genetic algorithm		Single-level genetic algorithm
Scenario	Best cost (10⁶ yuan)	Computation time (s)	Best cost (10⁶ yuan)	Computation time (s)
P = 30G = 100	0.893	82.59	0.932	68.17
P = 60G = 200	0.893	287.51	0.918	274.23
P = 100G = 300	0.893	730.87	0.893	674.60

Note: P = population; G = iteration parameter.

At the same time, it is evident that the computation time is directly proportional to the population size (P) and the number of iterations. Additionally, the computational time of the bi-level genetic algorithm is slightly longer than that of the single-level genetic algorithm under equivalent population size and iteration scenarios. However, the computational time in Table 9 represents a single run of the algorithm, while the optimal solution is the result obtained after 10 cycles of operation to obtain the best outcome. Therefore, when calculating for the same minimal optimal solution, the bi-level genetic algorithm takes approximately 826 s to obtain the optimal solution with P of 30 and G of 100. In contrast, the single-layer genetic algorithm requires about 6,746 s to achieve the optimal solution with P of 100 and G of 300. The solving efficiency of the bi-level genetic algorithm is eight times that of the single-level genetic algorithm.

Furthermore, observing the convergence iteration graph of the algorithm (Figure 9), it is evident that the bi-level genetic algorithm not only possesses a better minimum cost solution when P and G are relatively small, but also exhibits a significantly faster convergence speed compared with the single-level genetic algorithm. This is because the bi-level genetic algorithm can reduce the complexity and search space of the problem by decomposing it into two sub-problems, thereby enhancing the algorithm’s convergence speed. The rapid convergence rate demonstrates the advantageous nature of the bi-level genetic algorithm in solving optimization problems. By accelerating the convergence speed, the bi-level genetic algorithm can swiftly find better solutions and achieve superior results within a given time frame. Particularly for complex optimization problems, the rapid convergence speed of the bi-level genetic algorithm can substantially reduce computational time and resource consumption, thereby enhancing the efficiency and feasibility of the algorithm.

Finally, the stability of the solution of the bi-level genetic algorithm is much better than that of the single-level genetic algorithm. The two curves in Figure 10 connect the 10 optimal solutions of the bi-level genetic algorithm and single-level genetic algorithm algorithms, respectively. Observation of these curves shows that the curve of the bi-level genetic algorithm fluctuates less, and the probability of the solution reaching the minimum is 90%. In contrast, the curve of the single-level genetic algorithm fluctuates more, with only one out of 10 operations reaching the minimum. This indicates that the results of the bi-level genetic algorithm are more stable and make it easier to reach the optimal solution in multiple operations. It also implies that the bi-level genetic algorithm has better convergence and stability in solving problems. The results are more reliable compared with the single-level genetic algorithm.

Figure 10.

Diagram of algorithm calculation effect.

Large-Scale Calculator Effectiveness Tests

To enhance empirical validation and ensure the feasibility of applying the algorithm to larger-scale instances, we conducted experiments using a case containing 35 nodes, designed by Li and Sun ( 24 ). To contrast with the arithmetic results of the 10-node arithmetic example, in this study, we set the crossover probability to 0.9 and the mutation probability to 0.01, and ran it 10 times. We then compared the optimal results of the bi-level genetic algorithm and the single-level genetic algorithm under three different scenarios involving different P and G (Table 10).

Table 10.

Optimal Solution and Computation Time Under Different Scenarios

Scenarios	Bi-level genetic algorithm		Single-level genetic algorithm
Scenarios	Best cost (10⁶ yuan)	Computation time (seconds)	Best cost (10⁶ yuan)	Computation time (seconds)
P = 30G = 100	1.398	166.10	1.666	122.54
P = 60G = 200	1.351	548.61	1.529	469.61
P = 100G = 300	1.327	1,300.10	1.471	1,296.90

Note: P = population; G = iteration parameter.

From the perspective of algorithm performance, it is evident that the bi-level genetic algorithm exhibits a higher likelihood of generating superior solutions compared with the single-level genetic algorithm when employing identical parameter configurations. Even when P is set to 100 and G is set to 300, the optimal solution obtained by the single-level genetic algorithm remains higher than that achieved by the bi-level genetic algorithm using a P of 30 and G of 100. Clearly, the bi-level genetic algorithm outperforms the single-level genetic algorithm significantly in addressing large-scale computational cases, thereby providing a more viable solution to such problems.

It is important to note that the algorithm imposes greater demands on P and G when tackling large-scale computational cases in contrast to small-scale ones. Smaller population sizes and iteration parameters no longer enable the algorithm to converge toward the global optimum solution. Consequently, to pursue the attainment of an optimal solution, it becomes necessary to expand the settings of relevant parameters within the genetic algorithm. Simultaneously, the execution of large-scale computational cases substantially escalates the required solution time, and the amplification of P and G further compounds this effect, resulting in an exponential growth of the time required for the algorithm to achieve the optimal solution.

Therefore, although the bi-level genetic algorithm is far superior to the single-level genetic algorithm in solving large-scale algorithm cases and shows better solution feasibility, the limitations of genetic algorithms in solving large-scale algorithm cases still need to be addressed. Further research by scholars is still needed on how to further shorten the solution time of genetic algorithms in solving large-scale algorithm cases.

Impact of Time Penalty Strength and Time Windows

The hard time window penalty cost must be set to infinity given that the on-time delivery requirement must be completed. Therefore, the time penalty cost is mainly affected by $C_{e}$ (maximum holding cost per unit for early arrival) and $C_{l}$ (maximum penalty cost per unit for late arrival). For this reason, this paper sets the time penalty cost coefficient as a multiple of C_e and C_l as a determining factor affecting the strength of time penalty where the time penalty cost coefficient is denoted by $α$ . Meanwhile, to better observe the influence of the penalty cost coefficient on the decision making of multimodal carriers, this study sets two types of time window intervals, [40,70] and [20,30], which are relatively loose and relatively strict, as the constraints of the hard time window and the optimal service time as the control. The effect of time penalty cost coefficients on decision making is investigated under these two settings separately. The results are shown in Table 11.

Table 11.

Impact of Time Penalty Strength on Outcomes

$α$	Optimal route	Transport mode	Delivery time (hours)	Total operating costs (10⁶ yuan)	$\frac{Time penalty cost}{α}$ (yuan)	Total cost (10⁶ yuan)
[40,70][10,40,70,80]
1	on-time: O-2-6-9-Dgeneral: O-2-5-9-D	on-time: R-W-R-Rgeneral: W-W-W-W	on-time: 63.48general: 113.70	0.863	30,000	0.893
10	on-time: O-2-6-9-Dgeneral: O-4-6-7-D	on-time: R-W-R-Rgeneral: W-W-W-W	on-time: 63.48general: 116.00	0.863	30,000	1.163
15	on-time: O-4-5-9-Dgeneral: O-2-6-9-D	on-time: W-R-R-Rgeneral: R-W-R-R	on-time: 53.15general: 63.48	1.256	0	1.256
25	on-time: O-4-5-9-Dgeneral: O-2-6-9-D	on-time: W-R-R-Rgeneral: R-W-R-R	on-time: 53.15general: 63.48	1.256	0	1.256
[20,30][10,20,30,80]
1	on-time: O-4-6-9-Dgeneral: O-4-5-7-D	on-time: H-R-R-Rgeneral: W-W-W-W	on-time: 28.43general: 115.90	1.105	30,000	1.135
20	on-time: O-4-6-9-Dgeneral: O-2-5-7-D	on-time: H-R-R-Rgeneral: W-W-W-W	on-time: 28.43general: 116.10	1.105	30,000	1.705
30	on-time: O-4-6-9-Dgeneral: O-2-5-9-D	on-time: H-R-R-Rgeneral: R-R-R-R	on-time: 28.43general: 33.58	1.832	2,150	1.832
40	on-time: O-4-6-9-Dgeneral: O-2-5-9-D	on-time: H-R-R-Rgeneral: R-R-R-R	on-time: 28.43general: 33.58	1.832	2,150	1.918

Note: O = origin; D = destination; α = time penalty cost coefficient; H = highway; R = railway; W = waterway.

First, the time penalty cost coefficient has a significant impact on the transportation path and mode of transportation of general cargo. When the time penalty cost is small, the carrier will adopt the strategy of tolerating the loss from delayed arrival of general cargo and choose the waterway transportation mode which is time-consuming but relatively inexpensive. However, when the time penalty cost gradually increases, the carrier gradually cannot tolerate the loss from not arriving at the destination within the optimal service time. In this case, the carrier’s transportation strategy for general cargo gradually converges to on-time delivery of cargo to meet the need for cargo to arrive within the optimal service time.

Comparatively, observing the changes in the optimal paths and modes of transportation reveals that the combination of paths and modes of transportation is less affected by the time penalty cost coefficients because of the hard time window constraints on on-time delivery cargoes. There is a probability that the optimal decision of on-time delivery cargo has changed with the influence of the change in the decision for general cargo, however, the on-time delivery cargo is still certain to be delivered to the destination within the hard time window constraint. Moreover, by comparing the on-time delivery cargo transportation strategies under two different time window constraints, it is not difficult to find that the on-time delivery cargo transportation strategy is mainly affected by the time window setting. The closer the position of the time window interval is to zero, the more the on-time delivery strategy is interested in using faster highway and railway transportation.

On-time delivery cargo and general cargo may be viewed as two states of the same cargo. Consider general cargo with increasing time penalty cost coefficient as cargo with increasing time sensitivity. When the time penalty cost coefficient tends to infinity, this cargo becomes an on-time delivery cargo. Therefore, when the cargo time sensitivity is low, the cargo is insensitive to changes in the time window and the carrier tends to choose a low-cost strategy. When the cargo time sensitivity is high, the cargo is sensitive to time window changes and the carrier tends to choose a time-compliant strategy.

It is worth noting that the first two rows of data under both time constraints show that, as the time penalty cost coefficient increases, general cargoes take different transportation paths and the same mode of transportation, and the total operating costs incurred are the same. It is observed that the main reason for this phenomenon is because the case study is set to use box rates for waterborne transportation, so that the unit cost of waterborne transportation between any two nodes is the same. Therefore, under this setting, all the paths under the all-water transportation strategy consume the same transportation cost. It is reasonable that the two transportation paths in the Table 11 consume the same operating costs. This is not influenced by the time penalty cost factor.

In addition to this, the total multimodal transportation operating costs are increasing with the increase of the time penalty cost coefficient. This indicates that the increase in the level of time-sensitive services for cargo is inevitably accompanied by an increase in multimodal transportation operating costs. Correct evaluation of customer demand and adoption of appropriate strategies are essential for intermodal carriers to maximize their benefits.

Conclusions

In this paper, we study the container multimodal transportation route optimization problem considering the differences in time sensitivity of cargo, and set up a mixed time window penalty function to constrain the time sensitivity differences of different cargo transportation modes based on the differences of time sensitivity of just-in-time cargo and general cargo. An integer programming model was formulated with the objective of minimizing the total transportation cost. A bi-level genetic algorithm with the constraint of the mixed time window function was designed to search for a reasonable combination of multimodal transportation paths and modes. Finally, the bi-level genetic algorithm was implemented in MATLAB to solve the case study. The effectiveness of the bi-level genetic algorithm was further verified by algorithm comparison analysis and large-scale case testing. The results show that the algorithm is effective in dealing with mixed constraints, with advantages over single-level genetic algorithms. This indicates that, in the face of more complex problems, the appropriate choice of decomposing problem and using a multi-stage solution approach for the genetic algorithm solution can solve the shortcomings of the genetic algorithm in dealing with high-dimensional number problems to a certain extent. In addition, when analyzing the impact of time penalty strength and time window on decision-making outcomes, we observe that cargoes with higher time sensitivity are more likely to be affected by the time window setting. Decisions also tend to prioritize meeting time requirements at the expense of cost. Meanwhile, as the demand for time-sensitive services increases, operating costs rise accordingly. This means that services with stronger time constraints require more resources. Therefore, accurately assessing and categorizing the time sensitivity of customers can effectively prevent waste of resources in unnecessary areas, ultimately maximizing the outcome of decisions and ensuring the most efficient use of resources.

The study contributes to the existing research by filling the gap where few studies have constructed a mixed time window research focusing on the differences in time sensitivity of cargoes, thereby further improving the design system of mixed time window functions. It also helps to demonstrate the effectiveness of the bi-level genetic algorithm in solving multimodal transportation problems considering the different time sensitivity of cargo, thus enriching the application scenarios of bi-level genetic algorithms. In addition, the existence of differences in cargoes’ time sensitivity in transport practice will have a certain impact on the carrier’s transport decisions. Therefore, taking into account the current booming demand for customized services among consumers, this paper adds the need for on-time delivery cargo to the existing multimodal transport of general cargo, enabling carriers to determine the best transport strategy based on customer needs, cost, and other factors. Prioritizing sensitive cargo time window improves logistics service levels, helping carriers maintain quality customer resources and overall benefits ( 25 ).

The complexity of the multimodal transportation problem means that there are still many shortcomings in this study. In future research, factors such as the influence of different time sensitivities on customer satisfaction, finer segmentation of time sensitivity to be more in line with actual conditions, and methods of correctly identifying cargo time sensitivity and setting appropriate penalty costs can be incorporated.

Footnotes

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: Jinhai Yang, Danfeng Liang; data collection: Danfeng Liang, Zhuowu Zhang, Huanfang Wang; analysis and interpretation of results: Danfeng Liang, Hou Bin; draft manuscript preparation: Jinhai Yang, Danfeng Liang. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Social Science Foundation of China (grant No.19BGL177).

ORCID iD

Danfeng Liang

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