Optimization of multistage timeliness transit consolidation problem using adaptive-weighted genetic algorithm

Abstract

Cargo consolidation is becoming a crucial part of international transportation and changing the customer consumption patterns of the international community. Poor connections between different operations and the delay of international express have motivated sellers and logistics organizers to put timeliness first in international multimodal transport, especially during the COVID-19 epidemic. However, for cargo with small quality and multiple batches, designing an efficient consolidation network presents a set of unique challenges, including the coupling of multiple origins and destinations (ODs), and fully utilizing the capacity of the container. We defined a multistage timeliness transit consolidation problem to decouple the multiple ODs of the logistics resource. By solving this problem, we can increase the connectivity between different phases and make full use of the container. To make this systematic multistage transit consolidation more flexible, we proposed a two-stage adaptive-weighted genetic algorithm that mainly focuses on the edge area of the Pareto front space and the diversity of the population. Computational experiments indicate that the correlation between parameters has certain regular trends, and appropriate parameter settings can lead to more satisfactory results. We also confirm that the pandemic has a giant influence on the market share of different transportation modes. Moreover, the comparison with other approaches demonstrates the feasibility and effectiveness of the proposed method.

Introduction

The COVID-19 pandemic has accelerated the consolidation of cargo in the express industry, especially the integration of capacity. The gradual changes in consumer behavior brought about by COVID-19 have sped up the consolidation of commodities in all spheres of life, particularly in the express industry. The flattening and destocking reform in the fields of distribution, manufacturing, and circulation is slowing down as the epidemic gradually returns to normalcy, but consumers' high-frequency demand and fragmented orders have led to a rise of less-than-container-load (LCL) logistics; therefore, the cross-border shipping and turnover of items with small quantities and multiple batches will face serious challenges. Due to traffic control restrictions, cross-regional transport has become normal, which has increased labor and time costs and regional diversion costs. Port traffic also has been affected by the bullwhip effect of the highway, and the goods at the port have begun to be stranded; however, this phenomenon gave birth to the innovation of the port supervision model, which includes “linkage pickup and unloading” and “direct loading and lifting”. Railway transportation is relatively steady, orderly, and smooth, but there are issues during connections to highways and ports, and the utilization rate of comprehensive resources needs to be further improved.

Transit consolidation is a common logistics activity in express delivery, air freight and intermodal transport of containers; if a network lacks space–time synergy, inefficiency and underutilization of resources will become normal. The operation mode of transit consolidation is highly flexible and has unique advantages in cargo consolidation and container transshipment. The study of timeliness in transit consolidation networks in the context of multimodal transportation has unique reference value for transnational cargo transportation, especially during epidemics and pandemics. Narrow transit consolidation refers to cargo operations at a port terminal of a container freight station that involves only water transport. The generalized model considers not only the consolidation of goods but also the transshipment of containers between different modes of transportation.

In this paper, containers and cargo with different ODs will greatly increase the complexity of the problem. To better sort out the coupling of multiple ODs, we consider the timeliness of transit consolidation by dividing the whole network into several phases. The details are shown in Fig. 1. In this timeliness transit consolidation (TTC) system, four operation stages are given: first, we match the empty container and the truck; second, we match the cargo and the empty container; third, we match the heavy container and the vehicle; finally, we discharge the cargo at its destination. The second and third stages will be repeated many times during the operation of a container. The transport network factors include the nodes with different functions, the arc between the nodes that are passed through, different time windows, the logistics operation of cargo, containers, vehicles, and the scheduling of different vehicles to strike a balance among operating costs, time, and tonnage utilization.

Fig. 1.

An illustration of four operation modes of a container

The optimization of transit consolidation network is an extension of the capacitated vehicle routing problem (CVRP), which need to seek several routes for m number of vehicles with Q units’ homogenous capacity. The timeliness of transit consolidation is more about a vehicle routing problem with simultaneous pickups and deliveries with time window (VRP-SPDTW), which involves multiple origins and destinations (ODs) of goods that need to be distributed or collected to/from customers simultaneously and must meet the specified time window. For small batches of cargo with multiple ODs, it is of great significance to study the optimization of the timeliness transit consolidation problem (TTCP) in the case of multimodal transport. It involves multiple modes of operation for a container: the first matches the truck with the empty container, with the truck transported to the starting point of the goods to pick up, which also is the origin of the container; the second matches the container and the goods; the third matches the heavy container with the vehicle; and the fourth consists of arriving at the destination of the container, discharging all the cargo, and emptying the container. We know that the vehicle routing problem with simultaneous pick-up and deliveries (VRP-SPD) are NP-hard problems (Tasan & Gen, 2012), the problem of transit consolidation (TCP) is NP-hard (Lv & Yang, 2019), thus the TTCP is also NP-hard.

The remainder of the paper is organized as follows. Section 2 gives a summary of the study on relevant issues, and some of the main approaches we can used to solve the similar problems. Section 3 describes the problem of transit consolidation with double time windows in detail. Section 4 shows the details of the proposed algorithm. In this section, we mainly to improve the search ability of the solution space, the procedure of the algorithm and the thorough exploration of solution space are explained in detail. Section 5 shows the computational experiments and some comparative results of different combination of impact factors and algorithms. Section 6 presents the conclusions, the directions for further research.

Related work

Literature review

Scholars at home and abroad have studied the integration of transportation and time windows from different angles. However, there is no literature on combining transit consolidation and time windows, and that on the design of time windows is mainly about their upper and lower bounds. Few of the studies consider the continuity of the time between different logistics operations. The research mainly focuses on the construction of a time-window model and an effective algorithm solution. Demir (2016) provided a sampling average approximation method to consider the uncertainty of travel time. Aziez (2020) proposed a branch-and-cut algorithm to study the multi-pickup and delivery problem with time windows and presented the 2-index formulation and the asymmetric representative formulation. Yang (2016) analysed how to improve the overall efficiency by considering the time cost. These studies modeled and solved the multimodal transport problem of the entire container under certain conditions but did not consider the operation integration in the multiple origin and destination (OD) flow cargo transportation process of the container, that is, the optimization of the cargo during transport. Fazayeli (2018) considered multimodal distribution systems that combined warehouse location and multimodal transport together, and proposed a two-part genetic algorithm to get the best solutions. Guimares (2022) investigated the multicommodity flow problem containing multiple transport routes and time windows constrained by the inequality constraint between arrival time and displacement time. They examined and evaluated the performance of the three proposed heuristics for the proposed algorithms.

Consolidation is better for reducing costs and improving efficiency; for example, the implementation of urban consolidation centers (Aljohani & Thompson, 2020) or horizontal collaboration strategies (Ferrell et al., 2019) are two alternatives widely studied in the literature. However, in the relevant references to transit consolidation, most researchers have studied simultaneous delivery problems. The pickup and delivery problem with time windows (PDPTW) was first solved by Lu (2006), they proposed models and compared with a heuristic algorithm, considering the classical incremental distance measure in the insertion evaluation criteria and the cost of reducing the time window slack due to the insertion. Then the problem evolved into more complex simultaneous pickup and delivery problem with time windows (SPDPTW), scholars solved this NP-hard optimization problem in different ways, such as a co-evolution genetic algorithm with variants of the cheapest insertion method (Wang 2012), a parallel simulated annealing algorithm with a residual capacity and radial surcharge insertion-based heuristic (Wang 2015), an exact branch and price framework (Sun 2018). Györgyi (2019) extended the problem of pickup and delivery (PDP), considered its dynamics and stochastic. Park (2021) developed a genetic algorithm to solve the large-scale vehicle routing problem with SPDP.

At the same time, the TCP is a problem of resource integration, which is studied by the following scholars. Pérez (2018) used three families of valid inequalities with improved forcing constraints to establish the modal of express shipment service network. Estrada (2015) presented a methodology to identify when freight consolidation strategies are cost-efficient in the less-than-truckload carrier operations, which is helpful for our study. Hanbazazah (2019) established a mixed integer programming modal for a transportation problem with freight consolidation, which incorporated piecewise costs and delivery time windows. An information-sharing strategy is proved to be effective for approximating the whole PF space (Liu 2019). Hu (2018) considered the integration problem of multicommodity and combined the decision of commodity integration with the subproblem of inventory routing to achieve a critical transportation decision.

In research on consolidation transportation, only a few studies have combined transit and consolidation. Maknoon (2017) presented a mixed integer programming and a branch and bound algorithm to solve the less-than-truckload transshipment problem. Anoop (2022) provided a detailed study of freight consolidation in rail and road intermodal transport. Lv (2019) provided a detailed quantitative description of the transit consolidation process and established a comparison of the hub to illustrate the effectiveness of the transfer mode for small batches of cargo with multiple ODs, which proved that transit consolidation can effectively improve tonnage utilization. Moore (2023) consolidated cargo that have the same destination on the same day and integrated two complementary modes of transport, the transfer of cargoes from the road to other modes is achieved through mode splitting.

Table 1 shows an overview of the related literature. The differences are briefly presented in this section, including the problem features (simultaneous pickups and deliveries (SPD), container transit, cargo consolidation, and multicommodity), time-window types (continuity or time limit), and finally, solution methods (classified as exact or heuristic approaches).

Table 1.

An overview of the literature related to timeliness transit consolidation

Reference	Problem features				Time windows type		Solution approach
	SPD	Transit	Consolidation	Multi-Commodity	Continuity	Time Limit	Exact Method	Heuristic Method
Fazayeli et al. (2018)		✔			✔	✔		✔
Yang et al. (2016)		✔				✔		✔
Demir et al. (2016)		✔			✔	✔	✔
Guimares et al. (2022)				✔		✔		✔
Aziez, Côté and Coelho (2020)				✔	✔	✔	✔
Aljohani and Thompson (2020)			✔		–	–	–	–
Estrada and Robusté (2015)			✔			✔	✔
Györgyi and Kis (2019)	✔					✔		✔
Hanbazazah et al. (2019)			✔			✔	✔
Lu and Dessouky (2006)	✔					✔		✔
Lv et al. (2019)	✔	✔		✔	–	–		✔
Maknoon et al. (2017)		✔	✔		–	–	✔	✔
Park et al. (2021)	✔					✔		✔
Pérez et al. (2018)				✔		✔	✔
Sun et al. (2018)	✔				✔	✔	✔
Tasan and Gen (2012)	✔				–	–		✔
Hu et al. (2018)			✔	✔		✔	✔
Anoop and Panicker (2022)		✔	✔		–	–		✔
Moor et al. (2023)		✔	✔			✔		✔
Wang et al. (2015)	✔				✔	✔		✔
Wang and Chen (2012)	✔				✔	✔		✔
Total	8	7	7	5	6	16	8	13

– Means this information is not available in the paper

Most of the surveyed works address the CVRP and its variations by heuristic methods. This probably is due to the complexity of the problem when large, real-size instances are considered. Meanwhile, as an extension of CVRP, most of the literature considers the capacity of the vehicle or container. For the setting of time, only 6 studies focus on the continuity of the operation time, and most are about the upper and lower bounds of the time period.

Finally, it should be noted that among the 21 studies reported in Table 1, there is a large body of literature on the study of each feature of the problem individually but very little literature on the selection of multiple features for study; only 5 studies considered two or more features. It seems, therefore, that the study of TTCP has still been explored in a quite limited way.

Usually, the logistics operation is simplified into simple combinatorial optimization problems, and there are great limitations without consideration of the timeliness requirements in the logistics operation process. Moreover, there is very little literature that combines the integration of resources with the transit of containers and multi-OD of the cargo and container. Therefore, it is necessary to study the TTCP, and we designed the whole process, set the dual time windows of nodes and vehicles, and studied the impact of time-window conversion on the transit consolidation operation scheme.

The main contribution of this paper

To effectively address the multistage timeliness transit consolidation problem (MTTCP) with multiple objectives, a two-stage adaptive-weighted genetic algorithm (TSAWGA) is proposed; it is based on the mechanism of the nondominated sorting genetic algorithm-II (NSGA-II) combined with an adaptive genetic algorithm and adaptive-weighted fitness. The contributions and innovations of this paper are summarized as follows:

1. Problem. We address a new systematic timeliness transit consolidation that integrates simultaneous pickup and delivery (SPDP) with classical intermodal transportation to meet high-frequency demand and fragmented orders in the epidemic and pandemic period. Different combinations of the four stages sort out the coupling of multiple ODs and constitute the entire shipping process, simultaneous pickup and delivery exist in the process of cargo consolidation, and container transshipment also occurs according to the total costs. The timeliness factor is mainly about the seamless connections and time continuity of different processes.

2. Method. We contribute to the literature by presenting TSAWGA to improve the diversity and enhance the search capability of the whole solution space and the edge area of the Pareto front (PF) space. The improved genetic algorithm can avoid a local optimum of a single objective and take full advantage of the encoding and decoding design. In addition, the information of the optimum individual can be better inherited in the new individual through the discrete mutation operation.

3. Conclusions. The coupling of multiple origins and destinations can be well sorted in a multistage systematic transit consolidation problem, and the elasticity coefficient and fixed costs of containers can have regularity trends of influence on the results. We also demonstrate that COVID-19 has had a significant impact on the market share of transportation modes. Meanwhile, the proposed algorithm has absolute advantages in solving this problem and can be used in various transportation situations.

Problem description

Problem setting

The cargo transported by the container in the transit consolidation network has multiple ODs, as do the containers and vehicles, which have three dimensions of characteristics: physical characteristics (e.g., weight, volume, and package type), geographical characteristics (e.g., origin, destination), and commercial characteristics (e.g., price, quality, costs, and timeliness). Intermodal transportation can be achieved by a single participant or several participants who provide intermodal services, such as pickup, storage, customs clearance, import–export documentation, and exception handling. Typically, such supporting services are provided by freight forwarders who usually do not own ships or long-haul trucks.

To better illuminate these characteristics, there are some essential facilities (e.g., nodes, arcs, and vehicles) and some soft requirements (e.g., logistics organizers, time windows, quantities, and carbon emissions) that must be mentioned in this directed graph, in which the logistics organizer should clarify how, when and where the cargo, containers, and vehicles will be consolidated. The timeliness in the transit consolidation network is reflected not only in meeting the time windows of each node that is allowed to operate but also in selecting the optimal shift of delivery vehicles to minimize the waiting or delay times. Moreover, in this way, we can lower the social costs. When integrating logistics services, it is particularly important to balance and ensure the interests of the various participants, and the inevitable empty load problem that greatly increases social costs must be a primary consideration for cargo consolidation.

In a global logistics market with a single logistics organizer and multiple OD flows, transportation demands are specified by transportation means, container types and qualities (TEUs), time windows, origin–destination pairs and so on. We set up dual time windows for the different nodes (e.g., pickup nodes, transfer nodes, delivery nodes, consolidation nodes) and vehicles (e.g., trucks, ships, trains). Under the actions of the dual time windows, if the completion time of the container is longer than the departure time of the vehicle, there will be two options: wait or transfer to another mode of transport if one is available. The details are shown in Fig. 2.

Fig. 2.

Schematic diagram of space–time connection of a container in transit consolidation

The truck goes to the empty container yard at the beginning of a service in the entire transit consolidation, in which all the goods in the network reach their destination, and the services are finished. In Fig. 2, the second and third ways of logistics operation in Fig. 1 exist in cycles throughout the process. We assume the network is static, the vertical nodes are in the same area, and the truck begins to operate after receiving instructions to dispatch the container at $t_{\mathit{fs}}$. First, the truck arrives at container yard B from point A to load the empty container, then it loads three batches of goods; the total time is $T_o^c$, and the truck finishes the consolidation operation of the goods at starting point o, where $\left[{\mathit{ET}}_{1o},,,{\mathit{LT}}_{1o}\right]$ is the time window. $\left[{\mathit{ET}}_{2o}^{b_1},,,{\mathit{LT}}_{2o}^{b_1}\right]$ is in the time window of the ship $b_1$, ${{\mathit{ET}}_{2o}^{b_1}<T}_o^c<{\mathit{LT}}_{2o}^{b_1}$, then the container arrives at $m_1$ via $b_1$ and receives the fourth batch and discharges the first batch; in the time windows of $m_1$, the container operation completion time is still within the time window of the ship $b_1$, then continues to $e_1$, but since the completion time of $e_1$ is too long, or it has a different transport route than ship $b_1$, $b_2$ is selected to continue the transportation. The container also can use truck or railway to continue the transport, and the route of each cargo consists of the part or whole route of the container in which it is loaded. When all the goods reach their destination, the whole process of the service is complete.

Assumption

This problem contains the following assumptions:

1. The service object is a small batch of goods, and the supply place and the demand place are not located in the same country. Where devanning is allowed, it takes too long for a single node to achieve FCL (full container load) transportation.

2. In the whole process of transportation, there is only one logistics organizer in the process of global transportation of the same container. The owner can be different shippers from different countries, and the batches of the cargo are distinguished by the owner.

3. After the consolidation of the goods, the type of the container has been determined; only the same type of goods (e.g., fresh cargo cannot be loaded into ordinary containers) can be loaded together at the transit nodes, and the direction of the shipment depends on the cost trade-off or the most distant cargo that can be loaded before it is full.

4. The mandatory priority principle is the cargo that is loaded into the container must satisfy the quality constraints of the container. The highest priority of loading is that the cargo has the same destination as the goods that have already been loaded. The second priority is the cargo that can reduce the container’s empty carrying capacity or help to reduce total cost. The highest priority of discharging is the first time the container reaches the cargo’s destination.

5. A timer is set on the container to record the time consumed in each process. The container operation time at different nodes and the transit time at the same node are random, and they obey normal and uniform distributions, respectively. the container and the truck are seamlessly connected.

6. Different vehicles have different shifts, each with different time windows.

7. The cargo is sufficient, and the container never waits for cargo. We do not consider the insufficient demand problem in this transit consolidation network.

Mathematical formulation

Set and Indices
N	Set of nodes
K	Set of vehicles
F	Set of trucks
B	Set of ships
R	Set of trains
C	Set of containers
P	Set of products
$P_{m_i}$	A collection of goods that cannot be packed together with the cargo $m_i$
L	Set of routes
O	Set of intermodal terminals in the origin area
D	Set of intermodal terminals in the destination area
$l_{n_c}$	The route sequence of container c with $n_c$ nodes, which is an ordered combination of the operation nodes, which include origin, destination and transfer nodes, it starts from the first and ends with the last destination $l_{n_c}=\left\{n_1,,,n_2,,,\cdots ,,,n_z,,,\cdots ,,,n_{n_c}\right\}$
$E_c^i$	Set of nodes that the container c has passed in order when it reaches point i
i, j, z, h	Index
k, v	Vehicle
c	Container
p	Cargo
l	Route
o	Origin
d	Destination
Parameters
$c_{\mathit{ij}}^k$	Transportation cost per destination shipped by k from node i to node j (CNY/km)
$d_c$	Destination of the container c
$o_c$	Origin of the container c
$c_1$	Load cost per quality (CNY/kg)
$c_2$	Discharge cost per quality (CNY/kg)
$c_3^{\mathit{kf}}$	Transfer cost per container from transport mode k to mode f (CNY/times)
$F_c^k$	Fixed cost for container c under different transport k (CNY)
${\delta }_i$	Constant, if $i\in D_c$, the value is 1; if $i\in O_c$, the value is -1; others, the value is 0
$g_p$	The quality of the cargo p (kg)
${\mathrm{G}}_{\max }^{\mathrm{c}}$	Maximum load capacity of a container c (kg)
${\mathrm{G}}_{\min }^{\mathrm{c}}$	Minimum load cell of container c (kg)
${\eta }_1$	Coefficient of elasticity of load
$n_c$	The number of the cargo that loaded into the container c
$d_{\mathit{ij}}$	The distance between node i and node j (km)
$t_s$	The beginning time of service s (h)
$t_{\mathit{cs}}$	The beginning time of the container c in the service s (h)
$t_c$	The time that truck arrived at the container yard of container c (h)
$t_i^c$	The time that the container c arrived at node i (h)
$t_{\mathit{ij}}^k$	The transport time of k from node i to node j (h)
${\alpha }_i^c$	The time of heavy container c to wait for the vehicle after the operation of node i (h)
${\beta }_i^c$	The time of heavy container c to wait for the operation at node i (h)
$T_i^c$	Job completion time of heavy container c at node i (h)
$s_{\mathit{ij}}^k$	Average speed of k between node i and node j (km/h)
$q_i$	Penalty factor for node i
${\tau }_i^c$	Random operation time of container c to loading/discharging one batch of cargo at point i (h/batch)
${\phi }_i^c$	Random transfer time of container c at point i (h/times)
${\mathit{ET}}_{1i}$	The lower limit of the time window of the allowed job at node i (h)
${\mathit{LT}}_{1i}$	The upper limit of the time window of the allowed job at node i (h)
${\mathit{ET}}_{2i}^k$	The lower limit of the time window of the vehicle k at node i (h)
${\mathit{LT}}_{2i}^k$	The upper limit of the time window of the vehicle k at node i (h)
$c_4$	Unit time delay cost of empty container (CNY/h)
$c_5$	Unit time waiting cost of empty container (CNY/h)
$c_6$	Unit time waiting cost of heavy container (CNY/h)
$e_c$	Time cost of empty container c in the yard (including delay cost and waiting cost) (CNY/h)
n	The number of the containers that needed in the whole process (TEU)
Decision variables
$G_i^c$	The remaining bearing capacity of the container c after loading and discharging at node i (kg)
$X_i^{\mathit{kv}}$	Binary, the value is 1 if container c is transferred from transport mode k to the other v at node i, else the value is 0
$y_{\mathit{ij}}^c$	Binary, whether the arc (i,j) is travelled by the container c
$y_{\mathit{ij}}^k$	Binary, whether the arc (i,j) is travelled by the vehicle k
$x_{\mathit{pci}}$	Binary, whether the cargo p load into the container c at node i
$y_{\mathit{pci}}$	Binary, whether the cargo p is discharged from the container c at node i
${\gamma }_{\mathit{ij}}^{\mathit{pc}}$	Binary, whether the route of cargo p is in the container c and they have the same route at the arc (i,j)
$y_{\mathit{ij}}^{\mathit{ck}}$	Binary, whether the arc (i,j) is travelled by the container c using the vehicle k

Minimize

$$f_1\left(x_{\mathit{pci}},,,y_{\mathit{pci}},,,y_{\mathit{ij}}^k,,,y_{\mathit{ij}}^{\mathit{ck}},,,X_i^{\mathit{kv}}\right)=\sum _{k\in K}\sum _{i\in N}\sum _{j\in B_i}{c}_{\mathit{ij}}^k\times d_{\mathit{ij}}{\times y}_{\mathit{ij}}^k+\sum _{k\in K}\sum _{c\in C}\left(F_c^k,\times ,\sum _{i\in N},\sum _{j\in N},y_{\mathit{ij}}^{\mathit{ck}},/,\left(\left|N\right|,-,1\right)\right)+\sum _{c\in C}\sum _{i\in N}\sum _{p\in P}{c}_1\times g_p\times x_{\mathit{pci}}+\sum _{c\in C}\sum _{i\in N}\sum _{p\in P}{c}_2\times g_p\times y_{\mathit{pci}}+\sum _{i\in N}\sum _{k\in K}\sum _{v\in K}{c}_3^{\mathit{kv}}\times X_i^{\mathit{kv}}+\sum _c\left(e_c,+,\sum _{i\in E_c^i},\left(f,\left({\beta }_i^c,,,q_i\right),+,c_6,\times ,{\alpha }_i^c\right)\right)$$ 1

Minimize

$$f_2\left(y_{\mathit{ij}}^c,,,{\gamma }_{\mathit{ij}}^{\mathit{pc}}\right)=G_{\mathit{max}}^c\times \left(1,+,{\eta }_1\right)\times \sum _{i\in N}\sum _{j\in N}{d}_{\mathit{ij}}\times y_{\mathit{ij}}^c-\sum _{p\in P}\sum _{i\in N}\sum _{j\in N}{g}_p\times {\gamma }_{\mathit{ij}}^{\mathit{pc}}\times d_{\mathit{ij}}$$ 2

To meet the operational requirements of transit timeliness, maximizing efficiency with minimal cost is proposed. In this paper, we balanced the cost, time, and tonnage utilization to have a satisfactory MTTCP solution, simplified the main variables to make the model easier to understand, and did not consider only the volume of the cargo. Meanwhile, we built a linear model, especially the linearization of the objective functions and constraints. Objective (1) is to minimize the total cost related to loading, discharging, transferring, transporting, waiting times, and penalties. The first part is the transportation cost; the second part is the operation cost of the node, including the transit cost of the container and the loading and discharging cost of the cargo; the third part is the fixed cost of the container; and the fourth part is the time cost, including the waiting time cost and the penalty. This part ensures that each container can complete the consolidation and transportation within the time window of the node and the vehicle as much as possible.

Objective (2) is to minimize the invalid turnover of the cargo, which means that the real usage of the container should be largest and the unladen transport mileage should be smallest during transit consolidation. Described another way, the tonnage utilization of each container must be the largest. The first part is the loaded travel turnover, and the second part is the actual cargo turnover of the container. subject to

$$\sum _{i\in N}{y}_{\mathit{iz}}^c-\sum _{j\in N}{y}_{\mathit{zj}}^c={\delta }_z;\forall z\in N;\forall c\in C$$ 3

$$\sum _{c\in C}{y}_{\mathit{ij}}^c\le 1;\forall i\in N;\forall j\in N$$ 4

$$\sum _{k\in K}{y}_{\mathit{ij}}^k\le 1;\forall i\in N;\forall j\in N$$ 5

$$\sum _{c\in C}{y}_{\mathit{ij}}^c\ge \sum _{k\in K}{y}_{\mathit{ij}}^k;\forall i\in N;\forall j\in N$$ 6

$$\sum _{k\in K}\sum _{v\in K}{X}_i^{\mathit{kv}}=1;\forall i\in N;\forall c\in C$$ 7

$$y_{\mathit{ij}}^k+y_{\mathit{jz}}^v\ge 2X_j^{\mathit{kv}};\forall j\in N;\forall k\in K;\forall v\in K;\forall c\in C$$ 8

$$\sum _{c\in C}{x}_{\mathit{pci}}\le 1;\forall p\in P;\forall i\in N$$ 9

$$\sum _{c\in C}{y}_{\mathit{pci}}\le 1;\forall p\in P;\forall i\in N$$ 10

$$x_{\mathit{pci}}+y_{\mathit{pci}}\le 1;\forall p\in P;\forall i\in N;\forall c\in C$$ 11

$$\sum _{i\in N}\sum _{c\in C}{x}_{\mathit{pci}}=\sum _{i\in N}\sum _{c\in C}{y}_{\mathit{pci}};\forall p\in P$$ 12

$${\gamma }_{\mathit{ij}}^{\mathit{pc}}\le y_{\mathit{ij}}^c;\forall i\in N;\forall j\in N;\forall c\in C;\forall p\in P$$ 13

$$\sum _{c\in C}{\gamma }_{\mathit{ij}}^{\mathit{pc}}\le 1;\forall p\in P;\forall i\in N;\forall j\in N$$ 14

$$\sum _{k\in K}\sum _{j\in N}{y}_{\mathit{ij}}^{\mathit{ck}}\le 1;\forall i\in N;\forall c\in C$$ 15

$$\sum _{p\in P}{\gamma }_{\mathit{ij}}^{\mathit{pc}}\ge y_{\mathit{ij}}^c;\forall i\in N;\forall j\in N;\forall c\in C$$ 16

$$y_{\mathit{ij}}^{\mathit{ck}}\le y_{\mathit{ij}}^k;\forall i\in N;\forall j\in N;\forall k\in K$$ 17

$$y_{\mathit{ij}}^{\mathit{ck}}\le y_{\mathit{ij}}^c;\forall i\in N;\forall j\in N;\forall k\in K$$ 18

$$y_{\mathit{ij}}^c+y_{\mathit{ij}}^k-1\le y_{\mathit{ij}}^{\mathit{ck}};\forall i\in N;\forall j\in N;\forall c\in C;\forall k\in K$$ 19

$$G_i^c=\left(1,+,{\eta }_1\right)\times {\mathrm{G}}_{\max }^{\mathrm{c}}-\sum _{i\in E_c^i}\sum _{p\in P}{g}_p\times x_{\mathit{pci}}+\sum _{i\in E_c^i}\sum _{p\in P}{g}_p\times y_{\mathit{pci}};\forall i\in N;\forall c\in C$$ 20

$$G_{i-1}^c+\sum _{p\in P}{g}_p\times y_{\mathit{pci}}\ge \sum _{p\in P}{g}_p\times x_{\mathit{pci}};\forall i\in N;\forall c\in C$$ 21

$$t_c=t_s+d_{\mathit{ij}}^f/s_{\mathit{ij}}^f;\forall f\in F;\forall s\in S;\forall i\in N;\forall j\in O_c;\forall c\in C$$ 22

$$t_{\mathit{ij}}^k=d_{\mathit{ij}}^k/s_{\mathit{ij}}^k;\forall i\in N;\forall j\in N;\forall k\in K$$ 23

$$T_i^c=t_{\mathit{cs}}+\sum _{k\in K}\sum _{z=1}^{i-1}\sum _{j=1}^i{t}_{\mathit{zj}}^k\times y_{\mathit{zj}}^{\mathit{ck}}+\sum _{z=1}^i{\tau }_z^c\times \left(\sum _{p\in P},\left(x_{\mathit{pcz}},+,y_{\mathit{pcz}}\right)\right)+\sum _{z=1}^{i-1}\sum _{k\in K}\sum _{v\in K}{\phi }_z^c\times X_z^{\mathit{kv}}+\sum _{z=1}^{i-1}{\alpha }_z^c+\sum _{z=1}^i{\beta }_z^c;\forall c\in C;\forall s\in S;\forall i\in N$$ 24

$${\alpha }_i^c=L{T}_{2i}^k\times \sum _{j\in N}{y}_{\mathit{ij}}^{\mathit{ck}}-\left(T_i^c-⌊T_i^c/24⌋\times 24\right);\forall k\in K;\forall c\in C;\forall i\in N$$ 25

$$t_i^c=T_{i-1}^c+{\alpha }_{i-1}^c+t_{(i-1)i}^k\times y_{(i-1)i}^k+{\phi }_{i-1}^c\times X_{i-1}^{\mathit{kv}};\forall k\in K;\forall c\in C;\forall i\in N$$ 26

$${\beta }_i^c=\left\{\begin{array}{c}E{T}_{1i}-\left(t_i^c-⌊t_i^c/24⌋\times 24\right),t_i^c-⌊t_i^c/24⌋\times 24<E{T}_{1i}\hfill \\ 0,E{T}_{1i}<t_i^c-⌊t_i^c/24⌋\times 24<L{T}_{1i}\hfill \\ 24-\left(t_i^c-⌊t_i^c/24⌋\times 24\right)+E{T}_{1i},t_i^c-⌊t_i^c/24⌋\times 24>L{T}_{1i}\hfill \end{array}\right);\forall c\in C;i\in N$$ 27

$$e_c=\left\{\begin{array}{c}{(t}_{\mathit{cs}}-t_c)\times c_4,t_{\mathit{cs}}>t_c\\ {(t_c-t}_{\mathit{cs}})\times c_5,t_c>t_{\mathit{cs}}\end{array}\right);\forall c\in C;s\in S$$ 28

$$f\left({\beta }_i^c,,,q_i\right)=a{q}_i\times {\beta }_i^c+b;\forall c\in C;\forall i\in N$$ 29

$$x_{\mathit{pci}}\in \left\{0,1\right\};\forall i\in N;\forall c\in C;\forall p\in P$$ 30

$$y_{\mathit{pci}}\in \left\{0,1\right\};\forall i\in N;\forall c\in C;\forall p\in P$$ 31

$$y_{\mathit{ij}}^c\in \left\{0,1\right\};\forall i\in N;\forall j\in N;\forall c\in C$$ 32

$$y_{\mathit{ij}}^k\in \left\{0,1\right\};\forall i\in N;\forall j\in N;\forall k\in K$$ 33

$$X_i^{\mathit{kv}}\in \left\{0,1\right\};\forall i\in N;\forall k\in K;\forall v\in K$$ 34

$${\gamma }_{\mathit{ij}}^{\mathit{pc}}\in \left\{0,1\right\};\forall c\in C;\forall p\in P;\forall i\in N;\forall j\in N$$ 35

$$y_{\mathit{ij}}^{\mathit{ck}}\in \left\{0,1\right\};\forall i\in N;\forall j\in N;\forall k\in K;\forall c\in C$$ 36

$$0\le G_i^c\le \left(1,+,{\eta }_1\right)\times {\mathrm{G}}_{\max }^{\mathrm{c}};\forall i\in N;\forall c\in C$$ 37

Equation (3) is the flow balance constraint for containers. Constraints (4)–(5) mean that the random arc (i, j) is traveled by at most one container and one vehicle. Inequality (6) is the relationship between the decision variables. Constraint (7) ensures that only one mode of transport can be selected for the container to transfer. Inequality (8) is the continuity constraint of transport. Constraints (9)–(10) mean that the cargo can be loaded/discharged into/from at most only one container. Constraint (11) means that the cargo cannot be loaded and discharged simultaneously. However, the total number of loadings and discharges of one cargo is equal, as shown in Constraint (12). Constraint (13) is the relationship between the decision variables. Constraint (14) indicates that the cargo at most has the same route as one container on arc (i, j). Constraint (15) means that the container at node i can at most choose one vehicle and one arc to transport. Constraints (16)–(19) are the relationships between the decision variables. Constraints (20)–(21) are the loading capacity constraints of the cargo. Equations (22)–(26) indicate the truck arrival time at the container yard, transport time, operation completion time of the container at the node, waiting time for the vehicle while completing the operation at the node, and arrival time of container c at node i, respectively. The piecewise Eq. (27) indicates the waiting time of the container to operate at node i, and formula $t_i^c-⌊t_i^c/24⌋\times 24$ describes the time within the interval [0, 24]. Equation (28) is the waiting time cost of the heavy container to transport after the node operation is completed. Equation (29) indicates that a certain cost penalty is designed if it is not in the time windows, which is a linear function of the waiting time. If the container completes the operation early, it will cause a certain storage cost; 0 means the container completes the operation within the time window and satisfies the requirement, and there is no time cost to punish. If there is a delay and the arriving time is over the upper bound of the time window, this will cause a large waiting time, and the waiting time will be greater. The parameters a and b are nonnegative constants, and a is not equal to 0. The domains of the variables are defined in Eqs. (30)–(37).

Through the above target constraints, the total cost, including the transportation cost, fixed cost of the container, operation cost of the node, and transhipment cost of the container, must be minimized. The reasonable choice of transportation mode not only takes advantage of the low cost of waterways and railways but also makes full use of the high efficiency advantage of highways, minimizes the operation time, maximizes the utilization rate of container tonnage, and rationally arranges the transit consolidation network.

Solution approach

This MTTCP model is a linear mixed integer programming problem that includes two objectives, i.e., the lowest operation cost and the maximum utilization of the container, and there are two obvious conflicts between the objectives. On the one hand, as we all know, the transportation cost of the highway is the largest, but it can be more flexible than the others and its waiting time cost is the smallest; on the other hand, if a full container is shipped and all the cargo is shipped to the same destination, the tonnage utilization is 100%, but the waiting time cost to create a full container load may be the largest, so a balance must be found among these objectives.

We know that some transport problems can solved by branch cutting method and branch pricing method (Sun (2018); Aziez et al. (2020); Maknoon (2017)). However, with the increasing of complex and scale, intelligent algorithms are widely used (Wang et al. (2015); Park (2021); Lv (2019)). The MTTCP has multiple constraints and a larger scale of complex ODs, and it is an extended problem of pickup and delivery with a time window, which is an NP-hard problem.

In this paper, a method based on mixed integer linear programming is proposed. By considering special logistics operations and combining the points, vehicles, cargo, and containers to form the entire network, TSAWGA is used to solve the MTTCP, which includes the layout of the cargo and intermodal planning problem. The constraints above also increase the computational complexity, and the traditional method will have some infeasible solutions and greatly reduce the convergence speed of the algorithm, so we proposed TSAWGA. The first stage uses a priority-based encoding procedure proposed by Gen et al. (2018); the second stage determines the mode of transport between nodes. The decoding of the chromosome is divided into two parts: the first part is about two choices, if the cargo–container matches according to the mandatory priority principle proposed in the assumption, and the other the selection of the nodes through which the container will pass; the second part is the selection of the transport modes between the passing nodes. The fitness function is a constraint of the nondominated solution that we improve the interactive adaptive-weight genetic algorithm (iawGA) proposed by Gen, Cheng, and Lin (2008) to obtain the Pareto optimal solution. The probability of adaptive operators changes according to the fitness values of the iteration, and different parts use different genetic operators to reduce the possibility of infeasible solutions and improve the diversity of feasible solutions, which include crossover, mutation, reserve, and local search operators. In addition, we realized TSAWGA by the MATLAB solver.

The algorithm considers the individual needs of customers in the actual problem, sets node priorities for simultaneous container and cargo loading and unloading services, and uses double time windows for timeliness constraints, in which the optimal solution is formed. Since the solution space is relatively large, especially the distribution of transportation modes among different nodes, the optimal gene is retained by discrete mutation, which achieves information-sharing of the optimal gene among intergenerational populations while preserving the diversity of populations by crossover and reversal operations.

In essence, TSAWGA was selected to address this complexity problem for the following reasons:

1. The priority-based coding is well used for the MTTCP. With small batches of cargo with personality demands in a transit consolidation network, too many ODs exist, and a node must be visited at least once because it can be both the origin of one cargo and the destination of another cargo. The visiting sequence of the container and vehicle can significantly affect the MTTCP solutions. Using priority-based coding not only gives priority to the nodes being serviced but also gives priority to the cargo being loaded and discharged, which also can give a less strict order constraint.

2. The coding of the proposed TSAWGA is perfectly combined with the TTSP. The coding of TSAWGA accurately describes the variations and connections between these stages and seamlessly connects the four stages for simultaneous pickup and delivery during the transportation process. It connects two different stages by the only invariant (nodes) in the first stage, considering containers and vehicles as bridges to connect cargo from the origin to the destination and decoding to obtain the strategy of cargo layout and container transport under the constraints of capacity and time windows.

3. The solution space is fully searched using TSAWGA. The improved probability of crossover and mutation helps to maintain the diversity of the population, the local search operators provide a thorough search of the PF space, the mutation approach well preserves the optimal genes from the parents, and the refine strategy is the last line of defense to confirm the diversity of the population. All these factors make it possible to explore the neighborhood space of the problem and provide good exploration in searching for other local optimal solutions.

The improvement of genetic algorithm

We divided the chromosome into two parts: one regards the n node that is initialized by priority-based encoding, and the other regards the transport modes generated randomly. Different parts of the chromosome use different genetic operators. The first part is a partially mapped crossover (PMX) (Goldberg and Lingle, 1985), and the other stage is a two-point crossover (TPC) (James, 2007), which is used for the order of node assignment and order of transportation mode in the crossover strategy of TSAWGA. In addition, the mutation operation is generated by discrete mutation.

$$P_c=\left\{\begin{array}{c}\frac{k_1\left(f_{\mathit{max}}-f\right)}{f_{\mathit{max}}-f_{\mathit{avg}}},f\ge f_{\mathit{avg}}\hfill \\ k_3,f<f_{\mathit{avg}}\hfill \end{array}\right),P_m=\left\{\begin{array}{c}\frac{k_2\left(f_{\mathit{max}}-f\prime \right)}{f_{\mathit{max}}-f_{\mathit{avg}}},f\prime \ge f_{\mathit{avg}}\hfill \\ k_4,f\prime >f_{\mathit{avg}}\hfill \end{array}\right)$$ 38

Srinivas (1994) proposed a modified GA that adapts changes in the probability values of crossover and mutation based on the fitness values of the population. This algorithm can improve the convergence speed and accuracy of genetic evolution, effectively enhance its global search ability and avoid the problem of falling into local optimal solutions. The crossover probability $P_c$ and the mutation probability $P_m$ are determined according to the Eq. (38), $f_{\mathit{max}}$ denotes the maximum fitness of the population, $f_{\mathit{avg}}$ denotes the average fitness of the population, $f$ and $f^{{}^{\prime }}$ are the fitness of the individual after crossover and mutation, $k_i$ is a constant (i = 1, 2, 3, 4). If the probability is greater than the random number of [0,1], the crossover or mutation operation is performed. However, when the values of $f$ and $f^{{}^{\prime }}$ are close to $f_{\mathit{max}}$, the probability of $P_c$ and $P_m$ becomes very small, which causes the excellent individuals of the population to lose opportunities for evolution in the early stages of evolution.

Therefore, we use Eq. (39) to avoid premature convergence.

$$P_c=\left\{\begin{array}{c}\frac{P_{c1}}{\left(P_{c1}-P_{c2}\right)+e^{\frac{f_{\mathit{max}}-f}{f_{\mathit{max}}-f_{\mathit{avg}}}}},f\ge f_{\mathit{avg}}\hfill \\ k_1\ast P_{c1},f<f_{\mathit{avg}}\hfill \end{array},\right)P_m=\left\{\begin{array}{c}\frac{P_{m1}}{\left(P_{m1}-P_{m2}\right)+e^{\frac{f_{\mathit{max}}-f^{\prime }}{f_{\mathit{max}}-f_{\mathit{avg}}}}},f^{\prime }\ge f_{\mathit{avg}}\hfill \\ k_2\ast P_{m1},f^{\prime }<f_{\mathit{avg}}\hfill \end{array}\right)$$ 39

The whole procedure of the algorithm is shown in Fig. 3, and the details of the algorithm are shown in the remainder of this chapter.

Fig. 3.

The procedure of the TSAWGA

Encoding and decoding of the chromosome

The consolidation of the cargo and multimodal transport are the main logistic operations, so we divided the chromosome into two stages to better trace the cargo among the TC networks. In the real world, we always provide special services to fulfill the individual demand, so the first part is using priority-based chromosome encoding, which means that the goods whose origin or destination is at a higher priority node will be serviced first, that is, under the same operating environment, these special goods will be given priority for shipping or delivery.

The first part is randomly generated by a sequence of n nodes; the higher the number is, the higher the priority of the corresponding node to service. The second part generates sequences of 1, 2, and 3 randomly, and the number is $C_n^2$. The details of this encoding are shown in Fig. 4.

Fig. 4.

Encoding of the chromosome

The decoding of the chromosome includes two stages. In the first phase, according to the capacity of the container, the cargo is loaded into the container using the greedy strategy, and the transport mode is transferred into the matrix. The details are shown in Fig. 5.

Fig. 5.

Decoding to get the cargo number loaded into the container

The size of each cargo is shown in Table 2, and the total capacity of the container is 7. The route of the container is according to the orderly loading of the cargo and the information of its OD. For example, [5 2 4 3 1] is an ordered sequence of the first part of the chromosome, which means that [1 3 4 2 5] is the ordered nodes that passed. For the first container, it started at Node 1, where there are three batches of cargo; 4 is the largest, 1 has the same destination as 4, and the container still will have capacity after the loading of 1 and 4, so 2 also is loaded in this container. The next node is 3; according to the priority-based coding, it is the origin of 7 and the destination of 1 and 4, so after the operation at Node 3, the available capacity is 3, so 9 also should be loaded in.

Table 2.

The origin, destination, and the size of cargo

Cargo	1	2	3	4	5	6	7	8	9
O-D	1–3	1–5	5–1	1–3	5–4	2–4	3–5	4–1	4–3
Capacity	1	2	1	3	2	1	2	2	3

In the second phase, based on the first phase, we know the cargo loaded in each container. Then, according to the ODs of the cargo and the priority of each node that we generate while encoding the chromosome, we know the route of each container and complete the decoding of the chromosome. The second phase is shown in Fig. 6.

Fig. 6.

Decoding to get the route of each container

Taking Container 1 as an example, we know that the destination of the cargo that is loaded into the first container is 3, 5, 3 in sequence, but the priority of 5 is the lowest, so it is the last one to pass through in theory, while 5 is the origin, and it has the priority to be discharged, so 1, 3, 4, 3, 5 is the final sequence that Container 1 passes through. In the matrix, we know that Node 1 to Node 3 is through Transport Mode 3, Node 3 to Node 4 is through Mode 2, Node 4 to Node 3 is through Mode 2, and Node 3 to Node 5 also is through Mode 3. Through coding and decoding, we can obtain the route of each container to easily calculate the fitness value.

Genetic operators

We design different genetic operators to increase the diversity of the population and maintain information-sharing among different populations, which aims to enhance the search of the main PF area.

Crossover operators

We use the partial-mapped crossover (PMX) operator to act on the first stage. PMX refers to the random selection of two genes of the two parents as intersections and the location of gene exchange between two intersections, as the remaining gene segments generate two offspring according to the mapping relationship between the genes at the intersections. The second stage uses the two-point crossover, which exchanges the genes between the two points generated randomly. The crossover strategy is shown in Fig. 7.

Fig.7.

Crossover operation

Discrete mutation operators

In this section, we use a discrete mutation operation; one of the parents is randomly chosen from the nondominated solution of the previous iteration, while the other parent is the chromosome after the crossover operation.

In the first stage, if the random number between zero and one is greater than the mutation probability $P_m$, two pairs of cutting points are randomly selected, the segments between cutting points of the first parents are copied into the child directly, the duplicate elements are deleted from the second parent and the remaining genes are copied into the child in order, and then the first part of the offspring is generated perfectly, as shown in Fig. 8. For the second stage, Segments 4 and 5 are selected from the parents, Segment 6 is the difference between Segments 4 and 5, and Segment 7 is generated randomly between 0 and the differences. Then, the second part of the offspring is generated by Segment 4 plus Segment 7. By doing this, a new complete mutation individual with the label ‘Offspring’ is generated, making it possible for the genes of the parents to be well inherited in the offspring and enabling information-sharing across generations.

Fig.8.

Discrete Mutation operation

Reversal operator

To increase the diversity of the population, this paper uses the two-point inverse operator to act on the same stage, as shown in Fig. 9, and only the chromosome with high fitness can survive.

Fig.9.

Reversal operation

Local search of the solution space

We add local search operators to better utilize the solution space and give good exploration of local extreme value solutions, and individuals with good parental genes and their own traits produce a new population S. We first select the individuals with the top three rank values from the parent generation and then perform a local search operation to generate new individuals and add them to the new population.

For the multiobjective TTCP, two local search operations based on insert and exchange are defined as follows:

1. Change the sequence of nodes at the first stage by exchanging operations. Suppose nodes i and j are assigned to two different positions according to priority-based coding. This operation exchanges the priority of the cargo to be serviced at the two nodes to fulfill the time windows of the nodes and the vehicles. The exchange of the second stage is about the transfer and connection of the different vehicles.

2. Change the sequence of nodes at the first stage by inserting an operation. Suppose we randomly select a location m with a certain probability, remove node i at position m and insert it at another position l. As the sequence of the nodes changes, the cargo's serving sequence shifts to correspond with the information at node i. The same operation on the second stage changes the vehicles to be chosen to fulfill the demands.

The local search operations above explore the neighborhood space of node assignment and the sequence of orders. Although the operations are imposed on only one or two nodes, the sequence of demands at any node may be influenced.

Fitness calculation

Load the cargo at the node in sequence according to the priority and update the capacity of the container while passing through the node. The node with the highest priority is the starting point of the container. The destination of the loaded goods is the subsequent subset of the node, which is selected according to the fitness function, and the subsequent node with the smallest cost and largest tonnage utilization is reserved. In the timeliness transit consolidation network, there are several containers, each of which cycles in this way until all the demands are met. The balance between costs and tonnage utilization is constrained by the fitness function, which is proposed after we modified NSGA2 and the iawGA proposed by Gen, Cheng and Lin (2008), as shown in Eq. (40). $(f_1^{\mathit{min}},f_2^{\mathit{min}})$ and $(f_1^{\mathit{max}},f_2^{\mathit{max}})$ are the two extreme points of the MTTCP solution space.

$$e\left(v_i\right)=1/\left(\frac{\left(f_1^i-f_1^{\mathit{min}}\right)}{\left(f_1^{\mathit{max}}-f_1^{\mathit{min}}\right)}+\frac{\left(f_2^i-f_2^{\mathit{min}}\right)}{\left(f_2^{\mathit{max}}-f_2^{\mathit{min}}\right)}+p\left(v_i\right)\right)$$ 40

The parameters $1/\left(f_1^{\mathit{max}},-,f_1^{\mathit{min}}\right)$ and $1/\left(f_2^{\mathit{max}},-,f_2^{\mathit{min}}\right)$ in the Eq. (40) represent the different weights of the two objective functions. If individual i can be a dominated solution, there will be a penalty $p\left(v_i\right)=rank\left(v_i\right)/\max \left(r,a,n,k\right)$, which depends on its rank in the population. If i belongs to nondominated solutions, $p\left(v_i\right)=0$, and there is no penalty. Compared with the approach proposed by Gen, Cheng and Lin (2008), it will narrow the gap between suboptimal and optimal, better avoid becoming stuck in the local optimum and increase the diversity of the population. Individuals with lower fitness levels will be eliminated in this iteration.

Computational experiments and results

The cargo data are derived from the literature (Lv, 2019). The unit transportation cost between the nodes is randomly generated in different ranges. To satisfy the timeliness of the whole process, we design two different time windows that are independent, considering the time windows of each node and vehicle simultaneously. This is used to verify the effectiveness of the model and the algorithm.

In the international logistics network, every mode of transportation has its own origin, destination, and nodes that pass through at specific time zones, and there are many factors we need to consider. To simplify the complexity of the problem in this case study, we assume that each mode of transportation has 6 shifts available at each node and divide the time windows of the vehicles into several available time zones, as shown in Table 3. In addition, the transit time follows uniform distributions and handing time follows normal distributions, as shown in Table 4. The other input data are shown in Table 5, and the beginning time of the first container is zero.

Table 3.

Upper and lower limits of two time windows

Upper and lower limits	Time windows of each node or each vehicle
${\mathit{EK}}_1^n$	[0 4 6 7 12 17 0 0 4 6 7 12 17 0 0 4 6 7 12 17 0 0 4 6 7 12 17 0 0 4 6 7 12 17 0];
${\mathit{LK}}_1^n$	[6 15 12 18 20 23 3 6 15 12 18 20 23 3 6 15 12 18 20 23 3 6 15 12 18 20 23 3 6 15 12 18 20 23 3];
${\mathit{EK}}_{2i}^k$	[0 4 8 12 16 20 21; 1 5 9 13 17 21 0; 5 8 11 15 18 21 4];
${\mathit{LK}}_{2i}^k$	[2 6 10 14 18 22 24; 4 8 12 16 19 24 2; 7 10 14 17 20 23 6];

Table 4.

Transit time and conversion cost of unit containers under different modes of transportation

Modes	Random transfer time (h)	Random operate time at node (h)	Unit transfer cost of each container (CNY/times)
Highway	U (0.4, 0.05)	N (0.3, 0.05)	200
Railway	U (0.5, 0.1)	N (0.4, 0.1)	140
Waterway	U (0.6, 0.075)	N (0.3, 0.075)	100

Table 5.

Others input information

Penalty coefficient of time window $q_i$	N(0, 1)
Capacity of the container (kg)	26
Unit loading/unloading cost (CNY/kg)	15/10
Unit waiting cost $c_z$ (CNY/h)	100
Fixed cost of the container $F_c^k$ (CNY)	X11 = 2700; X22 = 3100; X33 = 2400

In this paper, the problem is solved in the MATLAB solver. The parameters of the time penalty function are a = 2, b = 0; the chromosome has a population size of 100, the parameters of the algorithm are setting according to the literature (Srinivas & Patnaik, 1994), generally, $k_1,k_2\in \left[0.5,,,1\right],P_{c1}=1,P_{c2}\in \left[0.5,,,1\right],P_{m1}=0.1,P_{m2}\in [0.05,0.1]$.

The effective of the model and the proposed algorithm

We can obtain the Pareto solutions of the MTTCP with two time windows by using TSAWGA, and the relatively optimal solution also can be achieved after running the code several times. To better analyze the results and increase the comparability of results, we change the description of the invalid turnover of the cargo into tonnage utilization, which is calculated using actual turnaround miles divided by the total turnaround miles at full load. During the iteration of the population with 100 individuals, the largest fitness value is 2.95 when the maximum iteration generation is 200, the overall cost is 46,615.60, and the average tonnage utilization is 70.63%; these are the relative best solutions of this problem. In addition, the PF number increased from 8 to 30, while the maximum number of iterations was increased. The overall trend was concave and gradually improved below 300 iterations, while for generations over 300, the number of fronts changed slightly, and the trend did not seem to change, which means that we should set the perfect iterations and parameters of the algorithms.

The solutions of this problem are shown in Tables 6 and 7. The results show that even if a container passes through more nodes or carries more cargo, it does not mean that it takes the most time to complete its service. For instance, Container 7 takes the shortest time, and Container 2 carries the most cargo and has the greatest tonnage utilization of 88.51%. Meanwhile, the waiting time will increase to balance the cost and time and make the greatest use of the container. The operation time of the node, the batch and capacity of the goods, the distance between the nodes, and the setting of the time windows are important factors that affect the solution.

Table 6.

The results of the transit and consolidation problem with time windows

No.	The order of the goods loaded into the container	The operating nodes in order	The way of transport	Tonnage utilization (%)
1	36, 19, 15, 18	21, 7, 19, 18, 14	2, 3, 1, 2	63.14
2	21, 25, 37, 22, 23, 39, 27, 24, 20	35, 16, 33, 6, 25	2, 3, 1, 1	88.51
3	5, 47, 32, 6	35, 20, 28, 34	3, 1, 2	84.63
4	42, 7, 9, 44, 13, 29, 38, 26	35, 22, 26, 32, 34, 31	2, 3, 3, 3, 2	63.02
5	50, 14, 11, 10, 30, 40, 28	35, 12, 2, 10, 13, 29	3, 2, 2, 2, 3	76.16
6	1, 41, 17, 45, 3, 31, 4	35, 12, 9, 3, 17	3, 3, 3, 3	69.59
7	43, 35, 16, 8, 33, 46	35, 22, 30, 23, 4, 27, 8	2, 2, 1, 2, 3, 1	59.83
8	48, 49, 12, 34, 2	15, 32, 11, 1, 24, 23, 10, 5	1, 1, 1, 2, 1, 1, 1	60.16
Ratio of different modes of transport 1, 2, 3			31.58%, 34.21%, 34.21%

1: waterway, 2: railway, 3: highway

Table 7.

The complete time of the transit and consolidation problem (h)

No.	Complete time at each node	Complete time of the containers (h)	Mean (h)	Standard deviation (h)
1	0.70, 3.42, 12.31, 31.30, 49.26	49.26	49.77	17.79
2	1.79, 4.53, 13.25, 20.66, 32.07	32.07
3	1.04, 17.32, 24.52, 41.85	41.85
4	1.35, 2.80, 13.56, 31.45, 43.53, 54.57	54.57
5	1.72, 12.16, 29.80, 33.98, 42.07, 54.57	54.57
6	1.80, 12.88, 28.27, 36.16, 54.97	54.97
7	0.78, 3, 4.52, 7.71, 9.93, 18.5, 26.26	26.26
8	0.31, 7.32, 17.03, 48.28, 54.42, 63.06, 79.87, 84.6	84.6

The completion times of different batches of goods from the origin to the destination are shown in Table 7. The variance in chromosomal encoding and decoding and the method of loading the cargo into the container result in a standard deviation of 17.79, which is significant variability, and the mean completion time is 49.77. The completion time of the container at the node it passes through depends not only on the amount of logistics operations at the node (such as more batches of goods need to be loaded into the container; container transit has occurred, etc.) but also on the operation time window of the node and the waiting time of the goods at previous nodes, so the node with high-priority operation has the right to complete the transportation first.

Different impact of loading elasticity coefficient and fixed cost

To verify the effect of different parameter settings on the results, we design different impact factors of the model, which are shown in Table 8. The fixed cost parameter is set so that the difference in fixed cost among the various modes of transportation increases in an arithmetic progression.

Table 8.

Impact factors design

Impact factors	Set of value
Loading elasticity coefficient	[0; 0.05; 0.07; 0.1; 0.15]
Fixed cost of container (CNY)	[2000, 2200, 1900; 2700, 3100, 2400; 3500, 4100, 3000]
EK1 (h)	[2 5 7 8 12 20 0 2 5 7 8 12 20 0 2 5 7 8 12 20 0 2 5 7 8 12 20 0 2 5 7 8 12 20 0]
LK1 (h)	[5 8 9 10 14 23 3 5 8 9 10 14 23 3 5 8 9 10 14 23 3 5 8 9 10 14 23 3 5 8 9 10 14 23 2]

To better illustrate the impact of different factors on the results of the model and discover the correlations between different parameters, we run the code several times with the same parameters and use the average Pareto solutions of each run as the data source for problem analysis. Figure 10 shows the relationship of the number of containers, tonnage utilization and costs under the condition of different loading elasticity coefficients and fixed costs. The following conclusions can be drawn from the results after conformal interpolation is used to build a fitted curve with the precise data of the special nodes.

Fig. 10.

Different fixed costs and loading elasticity coefficient on the results

Figure 10 shows the influence of several fixed cost settings on a single optimization result. It illustrates that the average number of required containers and cost have the same trend under the static freight network with the continuous increase of the loading elasticity coefficient, while the trend for tonnage utilization is the opposite increasing.

The number of containers is an important factor affecting the total cost. For cargo with small quality and multiple batches, the more containers used, the higher the cost and tonnage utilization will be, but the containers' empty rate also will increase. The PF solution set is to find a balance between these two objectives, so the downward trend of the total cost is elastic. When the coefficient is in the range of [0, 0.05] and [0.1, 0.15], the change is relatively flat, and the change in [0.05, 0.1] is relatively fast. As seen from the third plot of Fig. 10, the total costs are gradually reduced as the fixed cost decreases and the loading elasticity coefficient increases. The findings demonstrate that a modest fixed cost setting can effectively reduce the overall operation cost of timeliness transit consolidation.

The change in tonnage utilization rate is the opposite of the change in cost in theory; it is restricted by the packing rules and the balance of the overall cost. Therefore, when the elastic coefficient is [0, 0.05], the change is the largest, [0.05, 0.1] is the gentlest, and [0.1, 0.15] is relatively gentle. The findings demonstrate that modest capacity expansion can effectively reduce the number of containers required for the shipment of small batches of cargo when the overall freight network is static. However, as the capacity is gradually increased, there is little impact on cost and container utilization.

In general, [0.1, 0.15] is the best setting for the loading elasticity coefficient, which cannot be increased infinitely for safe transportation. The fixed cost setting of [2700, 3100, 2400] has an advantage in tonnage utilization over other settings, while [2000, 2200, 1900] has a definite advantage in total costs.

Impact of the pandemic on outcomes

COVID-19 mainly has influenced the freight and time window of nodes, and we calculate a certain percentage of influence for the different modes. The impact on the road is the greatest, with a proportion of 0.6, while the waterway is 0.3 and the railway is 0.1. These proportions reflect the impact on unit costs and the speed per hour of different modes of transport. Meanwhile, we shorten the time window of the nodes to avoid the unnecessary spread of the epidemic. The shortened time window is shown in Table 8, and EK1 and LK1 are the upper and lower limits, respectively.

The total cost of the solution with the highest fitness value is 45,569.25, and parts of the results are shown in Tables 9 and 10. The epidemic factor set in this paper will have an impact on only the cost and speed of different modes of transport, so the transport time between nodes will increase. In the case of the transport vehicle time window remaining unchanged while the node time windows are shortened, the time for goods waiting to be operated and containers waiting to be shipped will inevitably decrease. Meanwhile, the time not in the time windows will reduce greatly, which is greater than the transport cost, so the total cost will decrease compared to that in Chapter 5.1. The results show that the waiting time cost is reduced from 17,667.00 to 13,112.91.

Table 9.

The results of narrow time windows of nodes

No.	Complete time at each node (h)	Complete time of the containers (h)	Tonnage utilization (%)
1	1.66, 12.51, 24.42, 42.96	42.96	80.48
2	1.95, 8.39, 18.71, 26.28, 29.86, 32.68	32.68	65.03
3	1.55, 2.79, 6.86, 31.21, 55.33, 67.10	67.1	72.08
4	1.54, 14.39, 31.85	31.85	73.15
5	1.88, 5.83, 13.73, 21.11, 33.21	33.21	88.51
6	0.67, 6.81, 10.14, 26.53, 32.35, 44.17, 55.23, 77.36	77.36	68.47
7	2.79, 12.93, 21.87, 26.37	27.37	58.75
8	0.23, 12.47, 24.74, 28.81, 31.90, 56.54, 73.00	73.00	71.83
9	8.43, 36.76	36.76	45.79
Average		46.92	69.34
Standard deviation		19.79	12.28

Table 10.

The results of the transit and consolidation problem with narrow time windows

No.	The order of the goods loaded into the container	The operating nodes in order	The way of transport
1	47, 32, 6, 42, 7	35, 26, 28, 34	2, 2, 3
2	35, 16, 44, 10, 17, 13	35, 32, 27, 29, 9, 31	2, 2, 3, 2, 1
3	50, 9, 45, 14, 11, 40, 28	35, 22, 2, 10, 3, 13	2, 2, 2, 2, 2
4	3, 31, 30, 1, 41, 4	35, 12, 17	2, 1
5	21, 25, 37, 22, 23, 39, 27, 24, 20	35, 16, 33, 6, 25	2, 2, 1, 2
6	43, 5, 46, 49, 2	35, 30, 32, 1, 4, 20, 24, 23	3, 2, 2, 2, 2, 1, 3
7	8, 33, 29, 38, 26	22, 26, 34, 8	3, 3, 3
8	15, 18, 36, 19, 48	7, 19, 21, 15, 10, 18, 14	3, 1, 2, 3, 2, 1
9	12, 34	11, 5	3
Ratio of different modes of transport 1, 2, 3		16.67%, 55.55%, 27.78%

1: Waterway, 2: Railway, 3: Highway

In this paper, each container used is operated simultaneously; the smaller the number of containers used, the more time the container will take to complete the transport, so as the number of containers increases, the time to complete the transport decreases. As shown in Table 9, the average completion time of the container is slightly reduced to 46.92, but its standard deviation increases to 19.79. Meanwhile, the average tonnage utilization is 69.34%, which is significantly lower than that in the original time window, and the standard deviation is significantly higher to meet the slot of the narrow time windows.

Table 10 provides details on the packing layout and modes of transportation under the influence of the pandemic. It is obvious that the packing strategy will vary to reduce costs, and the mode of transportation will make a significant difference as well. Because railways are the most stable mode of transportation during the pandemic, the proportion of highways decreased from 34.21 to 27.78%, while railways increased to 55.55%, which is consistent with the actual situation.

In actual operations, the connection of different interfaces will not be as smooth and orderly as in the theoretical study. Although some of the conclusions are contrary to the impact of the actual epidemic on the transit consolidation problem, they also provide a reference for the time window setting and the unit cost setting of different transportation modes.

The evaluation of Pareto front

We use the reference solution set method to evaluate the Pareto outcome (Gen, Cheng, and Lin, 2008). We solve all the comparison algorithms using parameters with better performance and combine all the solutions to obtain the reference solution set, which will be better than any of the comparison algorithms.

The evaluation indicators include the number of pareto front $|S_p|$, the percentage of the non-dominate set while compared with the reference solution set $R_{\mathit{NDS}}(\left|S_p\right|)$, the distance between the non-dominate set and the reference solution set ${D1}_R$, besides, the value of objectives is also an indicator to prove the effective of the proposed algorithm. The calculation is shown in the Eq. (41)–(42):

$$R_{\mathit{NDS}}\left(\left|S_p\right|\right)=\frac{|S_p\in S^{\ast }|}{\left|S_p\right|}$$ 41

$S^{\ast }$ means the reference solution set, “| |” means the number of individuals in the solution set, $S_p$ is the non-dominate solution.

$$D{1}_R=\frac{1}{\left|S^{\ast }\right|}\times \sum _{i\in S^{\ast }}min\left(\sqrt{{\left(1/f_1\left(j\right)-1/f_2\left(i\right)\right)}^2+{\left(f_2\left(j\right)-f_2\left(i\right)\right)}^2}\right),j\in S_p$$ 42

Equation (42) is the average distance between the nondominant solution set and the reference solution set. We run the codes in the same parameters with 200 generations several times in different methods, and the relative best results are shown in Table 11.

Table 11.

Evaluation results based on the reference solution set

Measure	Methods
	TSAWGA	iawGA	NSGA2	spEA2
$|S_j|$	12	10	7	5
$R_{\mathit{NDS}}(\left|S_j\right|)$	0.75	0.4	0.429	0.4
${D1}_R$	22.44	99.70	43.86	16.26
CPU(times)	491.97	492.40	520.78	675.31

The results indicate that the proposed TSAWGA algorithm has a certain advantage over other algorithms; it has the highest percentage of individuals in the reference solution set, and it is obvious that TSAWGA has the shortest running time and its nondominant solution set is closer to the reference solution set. In summary, TSAWGA has an advantage over other algorithms in solving this problem.

The results of different algorithms are shown in Table 12. TSAWGA has an advantage in the cost and tonnage utilization of the MTTCP, especially the tonnage utilization, which proved the efficiency of the algorithm.

Table 12.

Two objective function values of different algorithms

Object		Algorithms
		TSAWGA	iawGA	NSGA2	spEA2
$f_1$	Min	0.6951	0.6800	0.6792	0.5899
	Max	0.7667	0.7495	0.7226	0.6732
	Average	0.7271	0.7194	0.7032	0.6416
$f_2$	Min	46,400.340	46,100.430	45,678.870	45,342.670
	Max	54,576.760	53,830.380	49,609.340	46,577.990
	Average	49,335.284	48,140.998	46,887.339	45,932.142

The conclusions above give us more consideration that subtle differences in parameter settings can lead to variations in solutions. For the organizers of transit consolidation, minimizing the overall cost and making full use of the container are their pursuits, so the elasticity coefficient of capacity [0.1, 0.15] is appropriate, with slight fluctuations in the results. It is important to weigh and consider various modes of transportation because the mode of transportation with the lowest price is not always the best option. Moreover, the influence of the pandemic can truly change the market share of different transportation modes, with railways being the most effective way to transport. In short, factors such as the priority of the services, the proportion of the transport modes, and the special needs of the customers should be considered in the timeliness transit consolidation.

Conclusions

In summary, from the perspective of the logistics organizer, this paper analyzes the timeliness factors that may exist in the process and establishes a multiobjective model to obtain a balance between the transport methods and the tonnage utilization of containers under limited resources. To better take advantage of each transport, the penalty function gives the delay and the penalty cost at the node in advance. TSAWGA was used to solve this complex problem. The fitness is to balance the objective to obtain the nondominant solution, and under the influence of the pandemic, special demands with priority will be met by using railways in transit consolidation, causing different transport-sharing strategies of the container capacity. Due to the constraint of the time penalty function, the delay or waiting will cause a large cost, and the trade-off between cost and time will increase the number of containers. The decision-maker can use the model and the algorithm we proposed to address the problems of TTC with time limitations and certain demands, especially considering the fixed cost of containers, the loading elasticity coefficient, and the time windows.

By analyzing the correlation between the loading elasticity coefficient and fixed cost, we know that within different ranges of the loading elasticity coefficient, the number of containers, the average tonnage utilization, and the total cost have different trends under the effect of different fixed costs. The pandemic is becoming normalized, especially causing a change in the transportation share market, and the proportion of railways is increasing. Although the costs and completion times have been reduced, the waiting costs have increased, tonnage utilization has shown a downward trend, and the proportion of highway transport has increased significantly. In addition, we evaluate the Pareto front solution by contrasting different algorithms. TSAWGA has an absolute advantage over the other algorithms mentioned in this paper that we can use to solve the problems of express delivery, air freight and intermodal transport of containers.

With the change in input, we can better use transit consolidation to provide a reference for transport organizers. Additionally, there are some factors that we have not considered. The setting of the shipping line of different vehicles and the dynamics of the demand in transit consolidation can be better studied in future research.

Declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.