Integrated Location Selection and Scheduling Problems for Inland Container Transportation

Abstract

Well-organized network configuration is the key to the success of inland container transportation systems. In this study, we firstly propose an integrated framework for the location selection of inland container depots (ICDs) and the scheduling of containers and trucks. The objective is to minimize the total cost of setting up the ICDs and transportation cost associated with trucks and containers. A mixed-integer linear programming (MILP) model is developed to solve the proposed problem. The computational studies show that the proposed decision approach is effective and can reduce the total operating costs of ICDs and transportation costs of containers. Sensitivity analysis on the impact of customer distributions and the number of ICDs on the total cost are conducted to reveal the characteristics of the problem. The utilization of ICDs can significantly improve the efficiency of the transportation network, i.e., the total cost is reduced by at least 27% for the proposed instances, and the transportation distance of empty containers is reduced by at least 4%. Finally, managerial insights and future research directions are provided.

1. Introduction

The imbalance between supply and demand in container transportation has led to the emergence of container transportation demand, including the reallocation of empty containers among ports and inland terminals, i.e., warehouses, depots, and railroad yards. Inland container transportation, as a continuation of sea transportation, is a key factor in ensuring the efficient flow of goods in and out of ports. The inland container transportation cost accounts for 20–80% of the total cost within the multi-modal transport context [1]. Optimizing the configuration of the inland transportation network and organizing empty- and/or full-container transportation services, as well as making full use of the well-developed comprehensive transportation system for inland transportation, are crucial for container transportation companies to ensure the quality of “door-to-door” container services. As a crucial hub of the inland transportation network, inland container depots (ICDs) connect terminal yards with hinterland customers. They function as a hub for distributing goods and storing empty containers. The establishment of ICDs in the inland container transportation network makes full use of the developed domestic highway transportation system, optimizes the inland transportation network, and enables rapid response to customer demands, while reducing truck transportation costs.

Global container throughput has increased from nearly 225 million TEUs in 2000 to 1.205 billion TEUs in 2020, with an average annual growth rate of 13% (World Shipping Council). The global container industry has consistently maintained a double-digit annual growth rate. However, due to the long-term imbalance in “East-West” and “North-South” trades globally, the utilization of containers remains low, and the disparity in industrial structures between domestic regions further exacerbates the imbalance of container supply and demand.

The demand for empty and/or full containers arises differently when the industrial division of labor results in disparities in the aggregation of import and export demands, differences in the quantity and types of import–export containers, and variations in the time required to fulfill demanded tasks. These factors directly contribute to the generation of container transportation demand. Inland container transportation, as the continuation of maritime shipping, plays a crucial role in ensuring the efficient flow of goods in and out of ports. Ref. [2] firstly proposed the inland container transportation problem and classified transportation demand into four types based on the OD and status (empty or full) of containers. Later, ref. [3] introduced time window restrictions in the container transportation context. Ref. [4] considered a variation in the inland container scheduling problem with different-sized containers. Recently, Ref. [5] considered drop-and-pull transport where the tractor and trailer are flexible and can be separated and combined. Ref. [6] further combined strip and discharge delivery of containers.

The superiority of container transportation can only be fully realized through intermodal transportation. This not only depends on the rational arrangement of maritime container shipping routes but also relies on the optimization and organization of “door-to-door” inland container transportation networks and container transportation services. A recent study by [1] firstly discussed the location selection problem with the consideration of empty container transportation between depots and shippers. Ref. [7] studied the inland container transportation network design with dry port and transport model selection. Ref. [8] later studied an inland dry port location problem and empty container relocation with volume discounts.

Based on our knowledge, studies dedicated to inland container depots as well as the vehicle–container scheduling problem in an integrated framework, are relatively fewer. This article intends to enrich the study of this field. The contribution of this study is twofold. Firstly, we propose an integrated framework for location selection of ICDs and container scheduling problems with the consideration of the transportation between all the entities in the inland transportation network, i.e., terminals, yards, ICDs, and customers. An inland container–truck scheduling model is developed to schedule the import and export of full and empty containers with the consideration of different ICD locations. The fixed costs of ICDs and variable costs associated with container transportation are simultaneously optimized. Secondly, we test the proposed model in extensive instances which are generated based on existing ones in the literature with additional attributes of customer distribution and time requirements. Managerial insights of the impact of ICD layouts and total costs under different scenarios are provided.

The rest of this article is arranged as follows: Section 2 provides a literature review. Section 3 introduces the research problem and a mixed-integer programming model. Section 4 provides numerical experiments for result analysis and sensitivity analysis, and Section 5 concludes the article with future research topics.

2. Literature Review

2.1. Inland Container Transportation and Scheduling

Existing research mainly focuses on container scheduling between ports, yards, and customer locations in the inland areas, with limited emphasis on ICDs. Ref. [9] using the example of the New York–New Jersey port area, demonstrated that the construction of inland empty container depots can effectively reduce the total system cost of empty truck mileage as well as empty container transportation. Ref. [10] studied the problems of container inbound transportation from customer locations to ports and scheduling with the objectives of minimizing transportation costs and reducing vehicle waiting time. Ref. [11] investigated the scheduling problem of container trucks between terminals and terminal yards, considering both the total waiting time of trucks and the emissions of all trucks while idling. Ref. [12] studied the inland container transportation problem in a hub-and-spoke model and proposed an intelligent model. Ref. [13] focused on a variant of the vehicle-routing problem with simultaneous pickup and delivery in the context of a land–seaport system for inland container transportation.

Due to the complexity of the problem, most approaches tend to use (meta)heuristics as the solution methodology. Several researchers refer to the inland container–vehicle scheduling problem as a container drayage problem (CDP). Ref. [14] address the passive and active means of transport in a routing problem and provide a large neighborhood search approach to solve the proposed problem. A branch-and-price-and-cut algorithm is proposed by [15] to solve the active–passive vehicle-routing problem. A cross-border container drayage problem is studied by [16]. An attribute decision model which simultaneously manages the driver, tractors, and trailers is presented. An adaptive labeling algorithm is presented. Ref. [17] studies a vehicle routing problem with trailers and transshipment (VRPTT). Two mixed-integer programming mathematical models and branch-and-cut algorithms are proposed to solve the problem. Ref. [18] study a variant of CDPs where tractors and trailers can be separated. A max-min ant colony optimization algorithm is provided to solve large-sized problems. Ref. [19] propose a tabu search algorithm to solve the problem. Ref. [20] study a CDP with container maintenance under separation mode. A branch-and-price-and-cut algorithm is developed to solve the proposed problem. Ref. [21] recently proposed a fuzzy complex proportional assessment technique to find the factors disturbing the sustainable performance of a shipping port.

2.2. Variations in Inland Container Transportation and Scheduling

Considering containers of different sizes in the aforementioned container scheduling problems has also received research attention. Ref. [4] studied the multi-size inland container transportation problem, where trucks can pick up at most two 20-foot containers or one 40-foot container along routes with different pickup and delivery locations. They formulated a mixed-integer programming model with the objectives of minimizing total travel distance and total truck operating time. Ref. [22] investigated the transportation problem of multi-size containers between ports and customer locations and proposed a container transportation scheduling model incorporating priority rules.

Incorporating empty container repositioning in the container transportation scheduling problem renders a more practical model. Ref. [2] studied truck scheduling for container transportation in a local area with multiple warehouses and terminals, considering four types of container movements: full inbound, full outbound, empty inbound, and empty outbound. Ref. [23] studied the container transportation scheduling problem with a focus on inbound transportation, considering constraints on customer coverage and terminal yard-time windows. They formulated a multiple traveling salesmen model with the objective of minimizing the total vehicle operation time, including waiting and transportation time. Ref. [24] studied the inland container transportation problem and developed a mathematical model with the objectives of minimizing truck travel time, considering vehicle routing, scheduling, and empty container repositioning. Ref. [3] investigated the container transportation problem between ports and customers, aiming to minimize the total truck travel time under time window constraints imposed by customers and terminals. Ref. [25] studied the scheduling problems of container inbound and outbound transportation between ports and customer locations and established a vehicle scheduling model with the objective of minimizing the transportation cost while satisfying customer time window constraints. Ref. [26] proposed a two-stage game model to optimally dispatch empty containers from inland container depots to terminals, deriving optimal delivery strategies for empty containers and the carrier’s optimal free dwell time. Ref. [27] presented a multi-commodity network flow problem that was able to optimize the fleet size of standard containers and combination containers while considering the scheduling of empty containers. Ref. [28] proposed two mixed-integer programming models for inland waterway linear transportation problems with empty containers repositioning.

Commercial software, i.e., Cplex, Gurobi, and Lingo, was used by [2,4,22,27,28], to solve the proposed transportation and scheduling problems. Due to the NP hardness of the problem, heuristic approaches have also been developed, i.e., the tabu search by [24], a two-stage heuristic by [3], and a hybrid tabu search by [25].

2.3. Integrated Location Selection and Container Scheduling

The attention of research to the joint optimization of ICD location and container transportation is relatively less. Ref. [1] firstly proposed the inland container transportation network design problem, considering container transportation path optimization. They studied the locations of inland depots under the premise of considering empty container scheduling, explicitly considering the uncertainty of empty container demand. They proposed a robust linear optimization model with the objectives of completing empty container scheduling within a certain time window and reducing total costs. Ref. [8] studied the dry port location problem with the consideration of empty container allocations. As for the solution methodology, ref. [1] proposed a branch-and-price algorithm with the consideration of only depots and shippers. Ref. [8] used Gurobi to solve generated instances.

In our study, compared with the existing research, we focus on container scheduling optimization in an integrated framework, considering transportation between terminals, yards, ICDs, and customers. The objective is to minimize the total setup costs and truck transportation costs. We also propose a mixed-integer programming model to optimize the empty container movement between terminals, ICDs, yards, and customer locations. The characteristics of our research compared with other works are summarized in

Table 1. Literature classification.

	Nodes	Empty Container	Solution Approach	Containers
[9]	C, Y, T	Y	Nonlinear programing	Homogeneous
[10]	C, T	-	Heuristic	Homogeneous
[2]	C, Y, T	Y	MILP	Homogeneous
[23]	C, Y, T	Y	MILP	Homogeneous
[25]	C, Y, T	Y	Hybrid tabu search	Homogeneous
[3]	C, Y, T	Y	Two-stage heuristic	Homogeneous
[24]	C, Y, T	Y	Tabu search	Homogeneous
[4]	C, ICD	-	MILP	Heterogeneous
[11]	Y, T	-	Simulation	Homogeneous
[26]	Y, ICD	Y	Two-stage game	Homogeneous
[12]	C, Y	-	Genetic algorithm	Homogeneous
[27]	Y, T	Y	MILP	Heterogeneous
[22]	C, T	-	LP	Heterogeneous
[13]	Y, T	-	Local search	Homogeneous
[1]	C, ICD	Y	Robust optimization	Homogeneous
[7]	C, Y, T	Y	MILP	Homogeneous
This paper	C, ICD, Y, T	Y	MILP	Homogeneous

Note: C: customer, ICD: inland container depot, Y: yard, T: terminal, -: not considered.

. By integrating the considerations of ICD location and container scheduling, these research efforts aim to improve the efficiency and cost-effectiveness of container transportation in the inland transportation system.

3. Problem Statement

3.1. Problem Description

The container transportation network studied in this article consists of four types of nodes: terminals, yards, internal container depots (ICDs), and customers (see Figure 1). Yards are located near terminals, while ICDs connect yards and customers, serving as hubs for cargo consolidation and empty container storage, and are normally located near customer clusters. Containers are transported between different nodes via trucks.

Customer transportation tasks include two categories: domestic transportation tasks and foreign trade transportation tasks. Domestic transportation refers to container transportation between customers in the inland areas, which is the IC (inland container) transportation task. Foreign trade transportation refers to round-trip transportation between terminals and customers. Terminals primarily handle import and export transportation tasks for foreign trade customers. In light of the task definitions by [2], based on the loading status of containers, foreign trade transportation tasks can be further divided into four types: IF (full-container import), OF (full-container export), IE (empty-container import), and OE (empty-container export), as described below:

(1)

IE (inbound empty container): starts from the terminal and should be transported to the yard, ICD, or delivery customers.

(2)

OE (outbound empty container): the empty container task has a unique destination, and the starting point is the yard, ICD, or any customer node.

(3)

IF (inbound full container): starts from the terminal, and the task destination is the delivery customer.

(4)

OF (outbound full container): starts and ends at unique locations, with the pickup customer as the starting point and the terminal as the destination.

The principal points and destinations of all the transportation tasks are summarized in

Table 2. Transportation tasks.

	IF	OF	IE	OE	IC
Principal point	Terminal	Pickup customer	Terminal	-	Customers/ICD
Destination	Delivery customer	Terminal	-	Terminal	Customers/ICD

. An illustration of container transportation is given in Figure 1. Terminals receive full and empty containers from pickup customers and send empty containers to delivery customers. Container yards send empty containers to pickup customers and terminals and receive empty containers from delivery customers and terminals. ICDs play a similar role to yards and transport any types of containers between terminals and customers.

The objective of the proposed problem is to minimize the total cost of truck transportation and ICD construction. Given the transportation tasks and time windows, the problem aims at fulfilling domestic and international transportation tasks with available trucks and containers and optimizing the repositioning of empty containers and truck scheduling.

For the sake of simplicity, we consider homogeneous vehicles to serve customer demands. Each vehicle can only move one container at a time, and all containers are the same size, i.e., 20 TEU. The ICD has unlimited capacity to accommodate a sufficient number of empty containers and trucks. Trucks start their routes from the ICD/yards and need to return to the ICD/yards after completing their transportation tasks. All transportation requests are known as a priori, and vehicles must complete all tasks within their given time windows.

3.2. Mathematical Model

3.2.1. Notations

• Parameters and sets:
• $s$: number of pickup customers;
• $r$: number of delivery customers;
• $sr’$: number of inland pickup customers;
• s$r’$: number of inland delivery customers:

  $$n=s+r$$

  $$n’=sr’+sr’$$

  $$m=n+n^{\prime }$$
• $E_i$: number of containers of IE;
• $E_o$: number of containers of OE;
• $d$: number of yards;
• $p$: number of ICDs:

  $$v=m+n+E_i+E_o$$
• $K=\left\{1,\dots ,k\right\}$: set of trucks;
• ${\partial }_1$: unitary travel cost of vehicles without containers;
• ${\partial }_2$: unitary travel cost of vehicles with containers;
• ${\partial }_3$: fixed cost of building an ICD;
• $t_{ij}$: travel time from node i to node j;
• $S_i$: service time at node i;
• $\left[a_i,b_i\right]$: time window of delivery at node i;
• $l$: time needed to load/unload a container;
• $p_i$: time needed to fill/empty a container at node i;
• $m_i$: initial number of vehicles at yard i;
• $o_i$: initial number of vehicles at ICD i;
• $d_k^{tru}$: staring point of vehicle k;
• $E_i^d:$ initial empty containers at yard i;
• $E^d$: initial empty containers at yards;
• $E_i^{icd}$: initial empty containers at ICD i;
• $E^{icd}$: initial empty containers at ICDs:

  $$V=V_T{\displaystyle \cup }{V}_D{\displaystyle \cup }{V}_{ICD}{\displaystyle \cup }{V}_C{\displaystyle \cup }{V}_{C’};$$
• $V_s=\left\{1,\dots ,s\right\}$: set of pickup customers;
• $V_r=\left\{s+1,\dots ,n\right\}$: set of delivery customers:

  $$V_C=V_s{\displaystyle \cup }{V}_r;$$
• $V_{s‘}=\left\{n+1,\dots ,n+sr’\right\}$: replications of pickup customers;
• $V_r=\left\{n+sr’+1,\dots ,n+n’\right\}$: replications of delivery customers:

  $$V_{C‘}=V_{s‘}{\displaystyle \cup }{V}_{r’}$$

  $$V_{OF}=\left\{m+1,\dots ,m+s\right\};$$

  $$V_{IF}=\left\{m+s+1,\dots ,m+n\right\};$$

  $$V_{OE}=\left\{m+n+1,\dots ,m+n+E_o\right\};$$

  $$V_{IE}=\left\{m+n+E_o+1,\dots ,m+n+E_o+E_i\right\});$$
• $V_T=V_{OF}{\displaystyle \cup }{V}_{IF}{\displaystyle \cup }{V}_{OE}{\displaystyle \cup }{V}_{IE}$: set of ports:

  $$v=m+n+E_i+E_o$$
• $V_{D_s}=\left\{v+1,\dots ,v+d\right\}$: set of starting yards;
• $V_{D_e}=\left\{v+d+1,\dots ,v+2d\right\}$: set of ending yards:

  $$V_D=V_{D_s}{\displaystyle \cup }{V}_{D_e};$$
• $V_{IC{D}_s}=\left\{v+2d+1,\dots ,v+2d+p\right\}$: set of starting ICDs;
• $V_{IC{D}_e}=\left\{v+2d+p+1,\dots ,v+2d+2p\right\}$: set of ending ICDs:

  $$V_{ICD}=V_{IC{D}_s}{\displaystyle \cup }{V}_{IC{D}_e}:$$
• $V=\left\{1,\dots ,v+2d+2p\right\}$: set of all nodes;
• $C=\left\{1,\dots ,r+E_i+E^d+E^{icd}\right\}$: set of all containers;
• $C_i^d$: empty containers at yard $i$:

  $${\displaystyle \sum }_{i\in V_{D_s}}{C}_i^d=\left\{r+E_i+1,\dots ,r+E_i+E^d\right\}$$
• $C_i^{icd}$: empty containers at ICD $i$:

  $$C^{icd}={\displaystyle \sum }_{i\in V_{IC{D}_s}}{C}_i^{icd}=\left\{r+E_i+E^d+1,\dots ,r+E_i+E^d+E^{icd}\right\}$$
• Decision variables:
• $T_{ik}$: time when vehicle k arrives at node i;
• $L_{ic}$: time when container c arrives at node i:

  $$x_{ijk}=\{\begin{array}{c}1 \mathrm{if} \mathrm{vehicle} k \mathrm{travels} \mathrm{from} i \mathrm{to} j\\ 0 \mathrm{otherwise} \end{array}$$

  $$y_{ijc}=\{\begin{array}{c}1 \mathrm{if} \mathrm{container} c \mathrm{is} \mathrm{delivered} \mathrm{from} \mathrm{node} i \mathrm{to} \mathrm{node} j\\ 0 \mathrm{otherwsie} \end{array}$$

  $$z_i=\{\begin{array}{c}1 \mathrm{if} ICD i \mathrm{is} \mathrm{open}\\ 0 \mathrm{otherwsie} \end{array}$$
• $T_{ik}$: time when vehicle k arrives at node i;
• $L_{ic}$: time when container c arrives at node i.

3.2.2. Parameter Setting

The node service time $S_i$ is related to the pickup and delivery locations of the container tasks. The service time consists of two parts: the time $p_i$ required to load/unload a container i and the time $l$ required to fill/empty a container. The calculation of service time $S_i$ is defined as shown in

Table 3. Calculation of service time $S_i$.

Type of Containers	Pickup Point	Delivery Point
Outbound full (OF)	$l+p_i+l$	$l$
Outbound empty (OE)	$l$	$l$
Inbound full (IF)	$l$	$l+p_i+l$
Inbound empty (IE)	$l$	$l$
Inland container (IC)	$l+p_i+l$	$l+p_i+l$

.

For example, for an export full-container task, the truck first needs to unload an empty container at the pickup customer. After completing the loading operation, the truck loads the full container and delivers it to the port. Therefore, the service time at the pickup and delivery points for an export full-container task can be represented as: $l+p_i+l$ and $l$, respectively.

In international trade transportation, when a truck performs two consecutive import/export container transportation tasks, it needs to visit the yard to unload/load empty containers. The corresponding travel time therefore includes the round-trip time to the yard and the time required for unloading/loading empty containers. The calculation for special cases of travel time $t_{ij}$ in international trade transportation is shown in

Table 4. Calculations of travel time $t_{ij}$ for external tasks.

Nodes	Travel Time
$i \in V_r{\displaystyle \cup }{V}_{IE}, j \in V_{IF}{\displaystyle \cup }{V}_{IE}$	$\underset{d \in V_D}{\min }\left(t(i,d\right)+t\left(d,j\right))+l$
$i \in V_{OF}{\displaystyle \cup }{V}_{OE}, j \in V_s{\displaystyle \cup }{V}_{OE}$	$\underset{d \in V_D}{\min }\left(t(i,d\right)+t\left(d,j\right))+l$

.

In mixed transportation involving both domestic and international trade, there are two cases where the truck needs to make additional visits to the yard/ICD:

(1)

If the truck’s previous transportation task was a domestic task and it has been assigned a new import task (IF/IE), it needs to unload empty containers at the nearest yard/ICD before executing the import task.

(2)

If the truck’s previous transportation task was an export task (OE/OF), after unloading the containers at the port, it needs to travel to the nearest yard/ICD to pick up empty containers and then start the next domestic transportation task.

The calculation for special cases of travel time $t_{ij}$ in mixed transportation tasks, considering the additional visits to the yard/ICD, is shown in

Table 5. Calculation of travel time $t_{ij}$ for internal tasks.

Nodes	Travel Time
$i\in V_{r\text{'}}, j \in V_{IF}{\displaystyle \cup }{V}_{IE}$	$\underset{d \in V_D{\displaystyle \cup }{V}_{ICD}}{\min }\left(t(i,d\right)+t\left(d,j\right))+l$
$i \in V_{OF}{\displaystyle \cup }{V}_{OE}, j \in V_{s\text{'}}$	$\underset{d \in V_D{\displaystyle \cup }{V}_{ICD}}{\min }\left(t(i,d\right)+t\left(d,j\right))+l$

.

3.2.3. Objective and Constraints

$$minz={\partial }_1\left({\displaystyle \sum }_{i\in V}{\displaystyle \sum }_{j\in V}{\displaystyle \sum }_{k\in K}{x}_{ijk}{t}_{ij}-{\displaystyle \sum }_{i\in V}{\displaystyle \sum }_{j\in V}{\displaystyle \sum }_{c\in C}{y}_{ijc}{t}_{ij}\right)+{\partial }_2{\displaystyle \sum }_{i\in V}{\displaystyle \sum }_{j\in V}{\displaystyle \sum }_{c\in C}{y}_{ijc}{t}_{ij}+{\partial }_3{\displaystyle \sum }_{i\in V_{IC{D}_s}}{z}_i$$

(1)

The objective function (1) represents the minimization of the total truck transportation cost and setup cost. The model constraints can be divided into three main parts: container flow-related constraints, vehicle routing-related constraints, and integrated constraints ensuring the coupling between vehicle flow and container flow.

• Container flow constraints:

$${\displaystyle \sum }_{j\in V}{\displaystyle \sum }_{c\in C}{y}_{ijc}=1 \forall i\in V_C{\displaystyle \cup }{V}_{C‘}{\displaystyle \cup }{V}_{IF}{\displaystyle \cup }{V}_{IE}$$

(2)

$${\displaystyle \sum }_{j\in V}{\displaystyle \sum }_{c\in C}{y}_{ijc}\le E_i^d,\forall i\in V_{D_s}$$

(3)

$${\displaystyle \sum }_{j\in V_{s’}}{\displaystyle \sum }_{c\in C}{y}_{ijc}\le z_i*E_i^{icd}, \forall i\in V_{IC{D}_s}$$

(4)

$${\displaystyle \sum }_{i\in V}{\displaystyle \sum }_{j\in V_{OF}{\displaystyle \cup }{V}_{OE}{\displaystyle \cup }{D}_e{\displaystyle \cup }IC{D}_e}{y}_{ijc}=1 \forall c\in C$$

(5)

$${\displaystyle \sum }_{i\in V_{OF}{\displaystyle \cup }{V}_{OE}{\displaystyle \cup }{D}_e{\displaystyle \cup }IC{D}_e}{\displaystyle \sum }_{j\in V}{y}_{ijc}=0 \forall c\in C$$

(6)

$${\displaystyle \sum }_{j\in V}{\displaystyle \sum }_{ i\in V_{D_s}{\displaystyle \cup } V_{IC{D}_s}{\displaystyle \cup }{V}_{IE}{\displaystyle \cup }{V}_{IF}}{y}_{ijc}=1 ,c\in C$$

(7)

$${\displaystyle \sum }_{j\in V}{\displaystyle \sum }_{ i\in V_{D_s}{\displaystyle \cup } V_{IC{D}_s}{\displaystyle \cup }{V}_{IE}{\displaystyle \cup }{V}_{IF}}{y}_{jic}=0 c\in C$$

(8)

$${\displaystyle \sum }_{j\in V}{y}_{ijc}=1 \forall i\in V_{D_s},c\in C_i^d$$

(9)

$${\displaystyle \sum }_{j\in V}{y}_{ijc}=z_i \forall i\in V_{IC{D}_s},c\in C_i^{icd}$$

(10)

$${\displaystyle \sum }_{j\in V_s{\displaystyle \cup }{V}_{D_e} }{y}_{ij\left(i-m-n-E_0+r\right)}=1 \forall i\in V_{IE}$$

(11)

$${\displaystyle \sum }_{i\in V_r{\displaystyle \cup }{V}_{D_s} }{\displaystyle \sum }_{c\in C}{y}_{ijc}=1 \forall j\in V_{OE}$$

(12)

$$y_{i\left(i-m\right)\left(i-m-s\right)}=1 \forall i\in V_{IF}$$

(13)

$${\displaystyle \sum }_{c\in C}{y}_{i\left(i+m\right)c}=1 \forall i\in V_s$$

(14)

$${\displaystyle \sum }_{c\in C^{icd}}{y}_{i\left(i+sr’\right)c}=1 \forall i\in V_{s^{\prime }}$$

(15)

$${\displaystyle \sum }_{j\in V}{y}_{jic}-{\displaystyle \sum }_{j\in V}{y}_{ijc}=0 \forall i\in V_C,c\in C$$

(16)

$${\displaystyle \sum }_{j\in V}{y}_{jic}-{\displaystyle \sum }_{j\in V}{y}_{ijc}=0 \forall i\in V_{C’},c\in C$$

(17)

$$L_{jc}\ge L_{ic}+t_{ij}+S_i-M\left(1-y_{ijc}\right) \forall i,j\in V, c\in C$$

(18)

Constraint (2) ensures that each customer node is visited exactly once, including the ports corresponding to IE and IF containers. Constraints (3) and (4) ensure that the number of empty containers originating from the yard/ICD does not exceed its capacity. Constraints (5) and (6) indicate that the destination of a container can be an export port, yard, or ICD. Constraints (7) and (8) specify the starting point of a container. Constraints (9) and (10) define the starting points of empty containers at the yard and ICD. Constraints (11) and (12) determine the possible pickup and delivery locations for the IE and OE transportation requests in the outer loop, while Constraints (13) and (14) define the starting points for the IF and OF transportation requests. Constraint (15) defines the starting point of the domestic transportation requests in the inner loop. The route and time continuity for containers are given by Constraints (16)–(18).

• Vehicle routing constraints:

$${\displaystyle \sum }_{j\in V}{\displaystyle \sum }_{k\in K}{x}_{ijk}=1 \forall i\in V_C{\displaystyle \cup }{V}_{C’}{\displaystyle \cup }{V}_T$$

(19)

$${\displaystyle \sum }_{j\in V}{x}_{\left(d_k^{tru}\right)jk}=1 \forall k\in K$$

(20)

$${\displaystyle \sum }_{i\in V}{\displaystyle \sum }_{j\in V_{D_e}{\displaystyle \cup }{V}_{IC{D}_e}}{x}_{ijk}=1 \forall k\in K$$

(21)

$${\displaystyle \sum }_{j\in V}{\displaystyle \sum }_{k\in K}{x}_{ijk}\le m_i \forall i\in V_{D_s}$$

(22)

$${\displaystyle \sum }_{j\in V}{\displaystyle \sum }_{k\in K}{x}_{ijk}\le o_i \forall i\in V_{IC{D}_s}$$

(23)

$${\displaystyle \sum }_{j\in V}{x}_{jik}-{\displaystyle \sum }_{j\in V}{x}_{ijk}=0 \forall i\in V_C{\displaystyle \cup }{V}_{C’}{\displaystyle \cup }{V}_T,k\in K$$

(24)

$$T_{jk}\ge T_{ik}+t_{ij}+S_i-M\left(1-x_{ijk}\right) \forall i,j\in V, k\in K$$

(25)

$$a_i\le T_{ik}\le b_i \forall i\in V,k\in K$$

(26)

Constraint (19) ensures that each customer and port is visited exactly once within a route. Constraint (20) defines the exact location of the vehicle’s starting point, while Constraint (21) ensures the uniqueness of the vehicle’s endpoint. Constraints (22) and (23) ensure that the number of vehicles originating from any yard/ICD does not exceed its capacity. Constraints (24) and (25) ensure the route and time continuity for vehicles. Constraint (26) defines the time window for vehicle node visits.

• Coupling of vehicle flow and container flow:

$${\displaystyle \sum }_{k\in K}{x}_{ijk}\ge y_{ijc} \forall i\in V_C{\displaystyle \cup }{V}_{C^{\prime }}{\displaystyle \cup }{V}_{IF}{\displaystyle \cup }{V}_{IE}{\displaystyle \cup }{V}_{D_s}{\displaystyle \cup }{V}_{IC{D}_s}, j\in V_C{\displaystyle \cup }{V}_{C^{\prime }}{\displaystyle \cup }{V}_{OF},c\in C$$

(27)

$${\displaystyle \sum }_{k\in K}{x}_{ijk}\ge y_{ijc} \forall i\in V_C{\displaystyle \cup }{V}_{C^{\prime }}{\displaystyle \cup }{V}_{IF}{\displaystyle \cup }{V}_{IE}, j\in V_{D_e}{\displaystyle \cup }{V}_{IC{D}_e},c\in C$$

(28)

$${\displaystyle \sum }_{k\in K}{x}_{ijk}\le y_{ilc} \forall i\in V_r{\displaystyle \cup }{V}_{r‘}{\displaystyle \cup }{V}_{IE}, j\in V_{IF}{\displaystyle \cup }{V}_{IE},l\in V_{D_e}{\displaystyle \cup }{V}_{IC{D}_e},c\in C$$

(29)

$${\displaystyle \sum }_{k\in K}{x}_{ijk}\le y_{ljc} \forall i\in V_{OF}{\displaystyle \cup }{V}_{OE}, j\in V_s{\displaystyle \cup }{V}_{s‘}{\displaystyle \cup }{V}_{OE},l\in V_{D_s}{\displaystyle \cup }{V}_{IC{D}_s},c\in C$$

(30)

$$T_{ik}=L_{ic } \forall i\in V_C{\displaystyle \cup }{V}_T, k\in K,c\in C$$

(31)

$$x_{ijk},y_{ijc},,z_i\in \left\{0,1\right\} \forall i,j\in V, k\in K,c\in C$$

(32)

$$T_{ik},L_{ic }: \mathrm{real} \mathrm{variables} \forall i\in V, k\in K,c\in C$$

(33)

Containers can only move through the coupling with vehicles, so Constraints (27) and (28) are introduced to ensure the interconnection between container flow and vehicle flow, where the vehicle flow may cover the container flow. Constraint (29) defines that when the next location visited by the vehicle does not require a container, the container flow ends at location $D^e$. Constraint (30) imposes that when the next location visited by the vehicle requires a container, a new container flow starts at location $D^s$. Constraint (31) ensures that when the vehicle moves a container, both the vehicle and the container leave the same node simultaneously. Constraints (32) and (33) are the constraints on the variables.

4. Computational Analysis

4.1. Instance Generation

We generate three types of instances based on different customer distribution scenarios: random distribution, uniform distribution, and clustered distribution. The illustration of the distributions is shown in Figure 2, where Ran, Uni, and Clu represent random, uniform, and clustered distributions, respectively. For each scenario, ten instances are generated. The generation of test instances follows the methods proposed by [2], and the numbers of international tasks are set to 25 and 35 for domestic tasks, for the sake of computational tractability. The customer points are randomly generated within a rectangular area, satisfying the requirements of the three distribution types. The number of trucks is set to 10. The number of candidate locations of ICDs is set to 12. Specifically, the K-means clustering method is employed to determine the positions of these candidates. Moreover, each instance has three sets of time window intervals: tight (0–1500), median (0–1900), and loose (0–2300). The time window value of each transportation task is generated from the interval based on a uniform distribution.

The model and instances are run on CPLEX (version 12.8) on a Lenovo ThinkStation P720, manufactured in Hefei, Anhui, China, with six Intel Core 3.4-G processors and 64 GB RAM, equipped with Windows 10 Enterprise. All the instances are run on CPLEX for 3600 s.

4.2. Impact Evaluation of ICDs

We firstly compare the total costs of the models with and without ICDs. The results of the model and instances without an ICD are acquired by imposing the number of functional ICDs to be zero. The results are summarized in

Table 6. Improvement by building ICDs.

Instance	NO. of ICDs	Total (%)	Dis. Full (%)	Dis. Empty (%)
Clu-Tig	8	27.41	32.30	4.57
Clu-Med	8	32.54	35.51	6.38
Clu-Los	8	35.72	38.26	8.11
Ran-Tig	10	33.14	35.93	4.45
Ran-Med	7	35.30	38.83	6.39
Ran-Los	8	37.41	40.12	9.42
Uni-Tig	12	37.73	40.07	4.58
Uni-Med	7	38.42	42.34	7.81
Uni-Los	8	41.24	44.34	9.03

. The first column is the instance class. Clu, Uni, and Ran represent the clustered distribution, uniform distribution, and random distribution of customers, respectively. Tig, Med, and Los stand for tight, median, and loose time windows, respectively. The second column records the number of functional ICDs in the final solution. The third column implies the percentage decrease in the total cost when ICDs are functional. The fourth and fifth column present the percentage decrease in the transportation distance of vehicles with a container and without a container, respectively. Note that the values of each entry in Table 6 are an average of ten instances. The numbers of ICDs are rounded to integers.

As we observe from Table 6, the necessity of building ICDs is obvious. The numbers of functional ICDs in all instances are positive, and functional ICDs can significantly reduce the total cost of the transportation and setup of ICDs. The total cost has an overall decrease of 35.43%. The values for full-container transportation distance and empty-container transportation distance are 38.63% and 6.75%, respectively. It should also be noted that the setup cost of ICDs has the negative effect of requiring more functional ICDs. For uniformly distributed customers, the number of functional ICDs is smaller than for the cluster distributed and randomly distributed customers. The reason is that clustered customers decrease the side effect of benefiting from a remote ICD. Additionally, tight time windows always render a larger number of functional ICDs than a loose time window. It is clear that having more functional ICDs gives more flexibility to meet customer time requirements. However, it also renders a longer distance of transportation and a larger setup cost, which can be observed from the last three columns in Table 6. Comparing the distances traveled by vehicles with containers, the loose time window has a larger decrease than the tight time window. From the last column of Table 6, we observe that the distance savings of vehicles without carrying a container are not significant compared with the total cost savings and distance of vehicles with a full/empty container. Due to the time window requirements, it might not be feasible to combine transportation tasks consecutively and assign tasks to one vehicle to finish.

To further examine the effect of ICDs on the total cost and diminish the side effects of the parameter values, we impose the number of functional ICDs as a constraint in the model and evaluate the results among different instance settings. The total cost is compared under each combination of the number of functional ICDs (ICD NO.), customer distribution, and time window constraints. The total cost consists of the total transportation cost and the fixed cost of the ICD. The results are presented in Figure 3 and Figure 4.

In Figure 3, we illustrate the changes in objective values among various situations. The blue bar represents the objective value of instances with cluster distribution, while green bar and red bar stand for uniform distribution and random distribution, respectively. The numbers above the bars are the numbers of ICDs considered as a constraint in the formulation. Note that the empty bars imply that the problems are not feasible. In the case that the number of ICDs is restricted to two, for most instances it is not feasible to finish the transportation tasks on time. We can also observe that the instances with randomly distributed customers have the largest total costs. This is mainly due to the higher transportation costs since the customers are located far away from each other. The instances with uniformly distributed customers have the lowest total cost. The reason is that the locations of the ICDs have the lowest average distance to most of the customers. As the number of ICDs increases, the total costs first decrease and then increase in most instances. The reason is the trade-off between the decreased transportation cost and increased set up cost.

Comparing the optimal solutions between instances with different numbers of ICDs, we observe that, for the time window constraint of 1500 (tight), the optimal numbers of ICDs for customer distributions close to uniform, random, and clustered distributions are 10, 10, and 8, respectively. It is obvious that for time-restrictive customers, more dedicated ICDs are needed to satisfy customers’ time window requirements. However, with only two ICDs, the existing inland container transportation network cannot meet the customer demands in the scenarios of random and clustered distributions. For the time window constraint of 1900, the optimal numbers of ICDs for customer distributions close to uniform, random, and clustered distributions are 12, 10, and 8, respectively. For the time window constraint of 2300, the optimal number of ICDs for customer distributions close to uniform, random, and clustered distributions is eight in all cases. In both cases, when there are only two ICDs, the existing inland container transportation network cannot meet the customer demands. The total costs of instances with randomly distributed customers are two times larger than the instances with cluster distribution and 2.3 times larger than the instances with uniform distribution.

4.3. Sensitivity Analysis

We further evaluate the relationship between the number of ICDs and the total cost for three different customer distribution scenarios and time windows. The results are summarized in Figure 5 and Figure 6.

In general, setting up ICDs can effectively reduce the total cost, and the total cost shows a trend of first decreasing and then increasing. As the number of ICDs increases, the total cost initially decreases. However, when the number of ICDs is set to be the optimal value, the additional benefits achieved by adding more ICDs become smaller compared to the fixed construction cost of ICDs, leading to an increasing trend in the total cost. Moreover, for the three types of customer distributions, the total cost is higher for smaller time windows.

From Figure 5 and Figure 6, it can be observed that under different time window constraints, as the number of ICDs increases, the total cost shows an overall decreasing trend when the customer distribution in the region is close to uniform, is random, or exhibits clustering characteristics. In the case of a random customer distribution for different time windows, most scenarios have no solution when the number of ICDs is less than three, meaning that it is not possible to satisfy customer demands within the required time through vehicle and container scheduling in the inland container transportation network. However, as the number of ICDs increases, solutions become available, indicating that setting an adequate number of ICDs in the inland container transportation network can meet customer demands, improve customer satisfaction, and reduce costs. In the case of a clustering or random customer distribution, setting up ICDs under smaller time window conditions can effectively improve the service capacity of the inland container transportation network and reduce the total cost. Furthermore, the reduction in total cost by setting up ICDs for a random customer distribution is the largest among all the instances.

Therefore, setting up functional ICDs in the inland container transportation network can achieve optimization of the total cost. Compared to other distribution patterns, when customer points exhibit random distribution and clustering, setting up ICD hubs can more effectively achieve cost reduction and efficiency improvement.

5. Conclusions

Inland container transportation, as the continuation of maritime shipping, is crucial for ensuring the efficient flow of goods in and out of ports. This study proposes an inland container transportation network with ICDs and investigates the scheduling problems of empty containers and truck dispatching in this network. A mixed-integer programming model is developed, and the model is validated through extensive instances. The results show that establishing ICDs in the inland container transportation network can effectively reduce the total cost and decrease the distance of empty truck runs.

We illustrate the effectiveness of the proposed model in extensive and representative instances. Based on the distribution characteristics of real customers, three types of scenarios are generated: average distribution, clustering distribution, and random distribution. The methods proposed by [2] are referenced to determine the customer scale and parameter values. The impact of different numbers of ICDs on the total cost, the relationship between the number of ICDs and the total cost under different scenarios, and the effects of ICD settings under different customer distribution scenarios are analyzed. Overall, in the existing inland container transportation network, the total cost can effectively be reduced, i.e., an overall 35.43% reduction in total cost and a 6.75% reduction in empty container transportation distance can be achieved. As the number of ICDs increases, the total cost shows a decreasing trend followed by an increasing trend. In the decreasing phase, the optimization effect on the total cost gradually diminishes, and when the number of ICDs reaches the optimal level, the fixed cost required for ICD construction exceeds the optimization effect of the total cost, resulting in an increase in the total cost. Additionally, with an increasing number of ICDs, the inland container transportation network can better serve customers within the required time range and improve customer satisfaction.

Based on the computational results, setting up ICD hubs in the existing inland container transportation network can make full use of the developed comprehensive transportation system in inland regions, improve the efficiency of inland container transportation, and reduce the overall construction and vehicle transportation costs of the network. The design of the transportation network should comply with the practical situations, i.e., customer distribution, time requirements, etc. For problems with randomly distributed customers, the total costs are more than two times larger than for problems with other distributions. For further research directions, it would be interesting to incorporate more practical restrictions in the model and develop (meta)heuristics to solve large-sized instances of the integrated problem.