Research on Optimization of International Multimodal Transportation Routes Based on Uncertainty Theory

Abstract

Delays in international multimodal marine transport can result in subsequent connection delays, leading to prolonged transportation time and increased uncertainty. To identify more reliable and cost-effective robust transportation routes, this study establishes a constrained planning model for multimodal transportation routes. Employing uncertainty theory, the model is transformed into a deterministic equivalent class model and ultimately incorporates an adaptive differential evolution (ADE) algorithm. Results of a case study indicate that: (i) the proposed model, compared with deterministic models, exhibits greater robustness and better aligns with practical transportation scenarios, resulting in substantial reductions in actual transportation time and cost; (ii) the solution efficiency of the ADE algorithm surpasses that of the genetic algorithm and Dijkstra algorithm; (iii) the start time of transportation and the confidence level of uncertainty also play crucial roles in influencing route selection. Therefore, decision-makers should consider a multifaceted approach when formulating transportation routes.

Keywords

planning and analysis transportation network modeling optimization public transportation planning and development multimodal network

International multimodal transport typically employs containers to seamlessly integrate traditional single modes of transport like water, land, and air transport. This integration connects various transport modes in series, facilitating the coherent transportation of goods from the point of origin to the destination ( 1 ). Because of the cost-effectiveness and efficiency of multimodal transport, it is progressively gaining significance in the international cargo transportation sector.

In previous studies on multimodal transportation route planning, the research objectives primarily focused on transportation time and cost. Some scholars approached the problem solely from the perspective of transportation cost, transforming the multimodal transportation route planning problem into a single-source and single-sink problem within the network. The emphasis was on reducing the number of transshipments during transportation ( 2 ), identifying shorter transportation routes ( 3 , 4 ), and minimizing the cost of container utilization ( 5 ). Additionally, certain scholars considered the external cost of transportation on the environment (6 –8). However, when shippers choose a route, they not only consider transportation cost, but also give priority to transportation time. Consequently, some scholars extended the research on route planning by addressing how to meet customer time requirements (9 –13). Additionally, some researchers directly investigated route planning models. These models aim to satisfy shippers’ dual demands for both time and cost simultaneously (14 –18). The solution algorithms for these models mainly focus on two categories: genetic algorithm (GA) ( 2 , 5 , 9 , 13 , 15 , 16 ) and Dijkstra algorithm (10 –12, 17 , 19 ).

Shippers’ requirements concerning delivery deadlines are sometimes ambiguous, however, meaning they seek the arrival of goods at the destination within a specified time window. In multimodal transportation route optimization, time window constraints can be categorized into two types: hard time windows and soft time windows ( 20 ). Hard time windows treat the time window as a constraint, considering the solution as infeasible if this condition is not met. On the other hand, soft time windows convert time into a cost, where failure to meet the time window constraints results in specific penalty cost ( 20 , 21 ). Whichever type of time window is used in research, most current transportation time handling methods rely on fixed values calculated based on distance and speed. Consequently, they adopt fuzzy mathematical programming methods to determine delivery deadlines and time windows. Nevertheless, these methods fail to provide a specific analysis of factors leading to transportation time uncertainty, in particular, the common issue of marine transport delays in the international multimodal transportation process.

Marine transport accounts for more than 75% of the total international trade volume and plays a critical role in international multimodal transport ( 22 , 23 ). International marine cargo transport comprises liner transport and chartering transport ( 24 , 25 ). Liner transport has fixed schedules, however, ships sailing at sea are inevitably subject to climate and natural conditions, which increase the likelihood of risks. Congestion is also a prevalent issue, frequently causing ships to be delayed, making it challenging to guarantee their punctuality. On the other hand, chartering has no predetermined port and route, meaning the date of the ship’s voyage cannot be estimated with accuracy ( 26 ). The latest maritime analysis report by Sea-Intelligence, a shipping analysis and consulting company, reveals that global container ship schedule reliability reached an all-time low in December 2021. Only one in ten ships arrived at their ports of call on time as planned, especially on the trans-Pacific route. The schedule reliability of container ships worldwide is reported to be 32%. In addition, Sea-Intelligence indicates that the average on-time rate of container ships for 2021 was 35.8%, the lowest recorded since Sea-Intelligence began tracking this indicator in 2011. On average, ships faced a delay of 6.86 days throughout the year ( 27 ). Rail and road transport are characterized by relatively fixed schedules, minimal susceptibility to weather disruptions, and a relatively low probability of delays. However, delays caused by marine transport may result in failure of synchronization with other modes of transport. This requires containers to be stored and awaiting departure, thereby making the overall transportation time of goods uncertain.

Subsequently, scholars recognized the coupled constraints between random demand and infrastructure construction or network design. Various methods emerged to address uncertainty, with the most widely applied being two-stage robust optimization models (28 –31). These models couple infrastructure and traffic expansion planning, achieving the coordinated optimization of investment and operational strategies. Others introduced a scenario generation approach to consider the uncertainty of traffic flow demand. Based on multiple scenarios, the steady-state distribution of traffic flow is characterized by the Wardrop user equilibrium principle. Corresponding equivalent constraints are derived and incorporated into the model ( 32 ). Moreover, some researchers proposed a novel multi-objective proactive distribution network planning model based on stochastic programming and uncertain random network (URN) theory ( 33 ). Others accounted for dual uncertainties in post-disaster transportation routes and times, presenting a feasible robust optimization model, where the dimension of uncertainty (number of traversed arcs) is implicitly deterministic ( 34 ). The solution algorithms for these models mainly involve heuristic algorithms or their variants, such as Benders decomposition algorithm (BDA) ( 31 ), adaptive large neighborhood search (ALNS) ( 28 , 35 ), and others.

We have organized and summarized some of the references most relevant to our research, as shown in Table 1. The results indicate that research in the field of multimodal transportation is primarily deterministic. Few studies addressing uncertainty in multimodal transportation focus on time windows. Unfortunately, the demand originates from the customers themselves and does not truly account for the uncertainty inherent in the transportation route. This is because of the oversight in quantifying uncertainty outcomes in specific scenarios. As a result, the current applicability of research related to time windows still requires further discussion ( 36 ). While the remaining studies consider the network’s impact, most of them are centered around single-mode transport. They are primarily related to the infrastructure construction and traffic planning of road transport. There is a fundamental difference between multimodal transport and single-mode transport. The multimodal transportation process involves numerous nodes, a longer process, and the presence of alternative transportation modes. Therefore, achieving the coordinated optimization of multimodal transportation routes and infrastructure construction is challenging. Considering the significant impact of marine transport delays on subsequent modes of transportation, and the scope of delay impact is often indeterminable, the abovementioned uncertainty handling methods may not be universally applicable.

Table 1.

Literature Exploring Different Aspects of Research Content and Model Algorithms

Reference	Application of scenarios	Objective function	Consideration of network or route impact	Transportation time	Methods for handling uncertain demands	Model-solving methods
Dib et al. ( 9 ) and Xiao et al. ( 13 )	MT	TT	N	D	NA	GA
Idri et al. ( 10 ) and Bożejko et al. ( 11 )	MT	TT	N	D	NA	Dijkstra
Gräbener et al. ( 12 )	MT	TT	N	D	NA	Dijkstra
Zhang et al. ( 15 )	MT	TC+TT	N	D	NA	GA
Lv et al. ( 2 ) and Zhao et al. ( 5 )	MT	TC	N	D	NA	GA
Bast et al. ( 17 )	MT	TC+TT	N	D	NA	Dijkstra
Sun et al. ( 37 )	MT	TT	N	U	TM	Cplex
Xu and Rong ( 16 )	MT	TC+TT	N	U	TM	GA
Demeyer et al. ( 19 )	MT	TT	N	U	ENE	Dijkstra
Tsang et al. ( 3 )	ROT	TC	Y	U	TM	GA
Ghilas et al. ( 28 )	ROT	TC	Y	U	TSRO	ALNS
Xie et al. ( 32 )	ROT	TC	Y	U	ENE	TDPLA
Xie et al. ( 29 )	ROT	TC	Y	U	TSRO	TLA
Xie et al. ( 33 )	ROT	TC	Y	U	URN	SOCP
Balcik and Yanıkoğlu ( 34 )	ROT	TT	Y	U	US	TSHA
Schiffer and Walther ( 35 )	ROT	TC	Y	U	RLRA	ALNS
Cadarso and Marín ( 30 )	RAT	TC	Y	U	TSRO	Cplex
Bagloee et al. ( 31 )	ROT	TC	Y	U	TSRO	BDA
This paper	MT	TC+TT	Y	U	UT	ADE

Note: MT = Multimodal transport; ROT = Road transport; RAT = Rail transport; TT = Transportation time; TC = Transportation cost; N = No; Y = Yes; D = Deterministic; U = Uncertain; NA = Not available: TM = Time window; ENE = Establishment of nonlinear equations; TSRO = Two-stage robust optimization; URN = Uncertain random network; US = Uncertainty set; RLRA = Robust location-routing approach; UT = Uncertainty theory; GA = Genetic algorithm; ALNS = Adaptive large neighborhood search; TDPLA = Three-dimensional piecewise linear approximation; SOCP = Second-order cone programming; TLA = Two-level algorithm; TSHA = Tabu search heuristic algorithm; BDA = Benders decomposition algorithm; ADE = Adaptive differential evolution algorithm.

In cases where there is a lack of sufficient samples or the range of influence cannot be effectively determined, an effective descriptive method is to consider such data as uncertain variables, and analyze them using uncertainty theory ( 38 ). Established in 2007 and continuously refined, uncertainty theory provides scholars with an axiomatic system. It is employed to characterize confidence and confidence levels, addressing uncertain variables in optimization models. This theory offers a solid mathematical foundation for effectively dealing with decision problems involving uncertainty, where obtaining precise information is challenging ( 38 ). To date, uncertainty theory has found widespread applications in various practical domains, including production planning ( 39 , 40 ), transportation (41 –45), solving functions with uncertain variables ( 46 ), storage theory ( 47 ), finance ( 48 ), route planning ( 49 ), uncertain statistics ( 50 ), and more. Uncertainty measurement fundamentally differs from probability measurement, and the essence of uncertain variables is distinct from random variables. Essentially, uncertainty measurement can be understood as the reliability of uncertain events occurring or the opportunity for uncertain target events to occur, rather than probability. It relies on the level of expertise about events and has a certain degree of subjectivity ( 38 , 40 ).

Based on the above analysis, we first investigated the transportation time from node $i$ to adjacent node $j$ , encompassing the expected time of delay under marine transport conditions. Subsequently, we investigated whether different modes of transport necessitate storage at transfer nodes, and if so, the corresponding duration required for storage. Finally, the total transportation time, encompassing transportation time from node $i$ to adjacent node $j$ , fixed transit time, and transit storage time, was determined. The opportunity-constrained planning approach was incorporated, aiming to treat the chance of meeting uncertain total transportation time requirements to achieve a specific confidence level, thereby obtaining more robust multimodal transportation schemes. Simultaneously, we considered various potential costs during transportation and formulated an optimization model for minimizing both total transportation time and cost in multimodal transportation routes. We also designed an adaptive differential evolution (ADE) algorithm to validate the effectiveness of the model and demonstrated the superiority of the ADE algorithm.

The remaining sections of this paper are structured as follows. The next section provides a formal description of the research problem and introduces several reasonable assumptions. The third section mathematically delineates a multimodal transport route optimization model that takes into account marine transport delay time. The fourth section presents the ADE algorithm we designed, tailored to the specific problem characteristics. To validate the feasibility and effectiveness of this approach, in the fifth section we present a case study. The final section presents pertinent conclusions.

Problem Statements and Assumptions

Problem Statement

The multimodal transportation network from node $o$ in one country to node $d$ in another country is illustrated in Figure 1. Figure 1 shows how the transportation process can be delineated into five segments: the initial phase involves overland transport from the point of dispatch $(o)$ to the point of departure (1, 2, or 3). Because of geographical constraints, the second phase from the departure point (1, 2, or 3) to node 4, 5 or 6 necessitates solely marine transport. The third and fourth phases comprise marine or overland transport from pertinent stations and ports. The final segment involves overland transport from the terminal node (10, 11, or 12) to the destination.

Figure 1.

International multimodal transport network.

Under the premise of disregarding other influencing factors, an optimal route has been identified in ideal conditions. The optimal route minimizes both the total transportation cost and total time from node $o$ to node $d$ in the multimodal transportation network: $o$ –1–6–8–10– $d$ . This is represented by the red line in Figure 1. The sequence of optimal transportation modes is: Road–Water–Rail–Water–Road. However, this transportation route may not necessarily be practically feasible. The reasons for this analysis are outlined below.

Rail and road transport are less affected by weather conditions, so the first segment of transportation can be considered relatively immune to delays. However, container multimodal transportation systems are susceptible to port congestion, often caused by unfavorable weather conditions. Consequently, ship arrival times become unpredictable, leading to frequent delays in marine transport. Therefore, there is a high probability of encountering delays during the second segment of the transportation process. Railway schedules are relatively fixed, there are no cases of trains waiting for delayed arrivals. Therefore, it becomes pivotal whether the container can arrive at node 6 in time to meet the departure time of the railway train. When delays persist, causing the total time for the container to arrive at node 6 plus the time needed for railway transshipment to exceed the departure time of earlier railway services, container storage becomes necessary because of the relatively fixed railway schedules. If the departure frequency at node 6 is low, for instance, some China–Europe freight trains operate only once a week, substantial storage time and cost will accumulate for the container at node 6. For this reason, any delay in marine transport has an impact on the selection of subsequent modes of transport. Marine transport introduces a high level of uncertainty into the overall transportation time.

The significant uncertainty in transportation time undoubtedly exposes shippers to potential losses. To adequately address varying demands of shippers concerning transportation time and cost, it is essential to conduct optimization studies on multimodal transportation route in cases of delays in marine transport. In situations where the specific scope of the impact caused by delays in marine transport cannot be accurately determined, an effective approach involves analyzing such data as uncertain variables. This paper is based on uncertainty theory methods, treating the total transportation time in multimodal transport as an uncertain variable, establishing mathematical optimization models for analysis.

Model Assumptions

According to the actual situation of international multimodal transport, to simplify the research, we make the following reasonable assumptions about the problem:

Air transport is rarely used in bulk container multimodal transport, so this paper does not take air transport into consideration.

The goods cannot be disassembled or consolidated during transportation, and their integrity should be maintained.

The transformation of the transportation method can only occur at the transit node, and each transit node can only change one transportation method.

Capacity constraints of the nodes are ignored.

The distribution of total transportation time uncertainty in multimodal transport follows a zigzag distribution.

The packing method of the container is Full Container Load (FCL).

We consider only conventional delays caused by marine transport, excluding other extreme delay scenarios, such as when the shipper’s containers are dumped by the container liner.

Mathematical Formulation

Description of Model Symbols

We have studied the time and cost of containers during intermodal transport, with full consideration of the delays occurring in marine transport. To understand the international intermodal transport route optimization model better, the symbols and explanations used in the formula are shown in Table 2.

Table 2.

Sets, Parameters, and Variables

Sets and indexes
$(W \cup R \cup H)$	Set of transport modes, where $W$ is marine transport, $R$ is rail transport, $H$ is road transport.
$N$	Set of nodes in the multimodal transport network.
$o, d$	The origin and destination of intermodal transport, $o, d \in N$ .
$i, j, m$	Index of intermediate node station, $i < j < m$ and $i, j, m \in N$ .
$n, l$	Index of the modes of transport, including road, rail, marine transport, $n, l \in$ (W ∪ R ∪ H) .
$a, b$	Index of the ath/bth trains/cars/ships departing from the station.
Parameters
$d_{ij}^{n}$	Transport distance from node $i$ to node $j$ using transport mode $n$ .
$v^{n}$	Average transportation speed of the transport mode $n$ .
$λ_{ij}$ , $γ_{ij}$	Parameter of the probability density function of ship delay between node $i$ and node $j$ .
$ψ_{ij}$ , $ω_{ij}$	The number of late ships and the total number of ships sailing from node $i$ to node $j$ , respectively, while $ω_{ij}$ is the sum of the number of ships, including both delayed to port $ψ_{ij}$ and not experiencing delays.
$L_{ij}^{l, a}$	Actual departure time when the container is transported from node $i$ to node $j$ by the ath train of the lth transport mode.
$φ_{jm}^{l, a}$ , $ξ_{jm}^{l, a}$	The earliest and latest allowable arrival time at the station for the ath shift sent by transport mode $l$ from node $j$ to node $m$ , respectively.
$A_{j}$	The moment when the containers arrive at node $j$ .
$T_{o}$	Departure time of the containers at the place of origin.
$t_{j}^{nl}$	Transit time per unit for converting transport mode $n$ to $l$ at node $j$ .
$s_{ij}^{n 1}$ , $s_{ij}^{n 2}$	The corresponding basic freight rate 1 and basic freight rate 2 for rail or road transport, respectively.
$s_{ij}^{n 3}$	Basic rate of container from node $i$ to node $j$ using marine transport.
$q$	The billable quantity of the container.
$p$	The volume of goods transported.
$ρ_{j}$	The unit storage fee generated at node $j$ .
Variables
$\partial_{ij}^{n}$	Theoretical transportation time required to travel from node $i$ to adjacent node $j$ using transport mode $n$ .
$f_{ij}^{n}$	Probability density of ship delays between node $i$ and node $j$ .
$g_{ij}$ , $g_{ij}^{0}$	Probability of a ship being delayed and not be delayed between node $i$ and node $j$ , respectively.
$E_{ij}^{n}$	Expectation of late time occurring from node $i$ to node $j$ using marine transport, $n \in W$ .
$T_{ij}^{n}$	Transportation time for various modes of transport from node $i$ to adjacent node $j$ .
$t_{j}^{c}$	The charge storage time generated at node $j$ .
$t_{j}^{t}$ , $t_{j}^{f}$	The fixed transit time and free storage time at node $j$ , respectively.
$T_{od}$ , $C_{od}$	The total transportation time and cost, respectively, in case of delays in the multimodal transportation process.
$η_{od}$	Actual total transportation time of multimodal transport—an uncertain variable that obeys a certain distribution.
$Z (a_{od}, b_{od}, c_{od})$	A zigzag uncertain variable for the total multimodal transportation time from origin to destination.
$α_{od}$	The transportation time for the planned route meets the uncertainty level set for the actual arrival time.
$S_{ij}^{n}$	The transportation cost required to travel from node $i$ to node $j$ using transportation mode $n$ .
$G_{ij}^{n}$	Container surcharge from node $i$ to node $j$ using the nth mode of transportation.
$C_{j}^{nl}$	Unit transport cost of converting transport mode $n$ to $l$ at node $j$ .
$δ_{j}$ , $Q_{j}$	The total transit cost and storage cost of containers at node $j$ .
$T_{od}^{\max}$ , $C_{od}^{\max}$	The maximum total transportation time and cost that the shipper can bear, respectively.
$x_{ij}^{n}$	Indicates if the plan decides to choose transport mode $n$ from node $i$ to node $j$ (1), or not (0).
$y_{j}^{nl}$	Indicates if the scheme decides to switch the transportation mode $n$ to $l$ at node $j$ (1), or not (0).

Related Time for Multimodal Container Transport

In the process of international container intermodal transportation, the total transportation time mainly consists of three parts: transportation time from node $i$ to adjacent node $j$ , fixed transit time, and transit storage time.

1. Transportation time from node $i$ to adjacent node $j$

When a container is transported from node $i$ to its adjacent node $j$ , using rail or road transport, it can be assumed that there is no delay in the transportation process, as these modes are less affected by weather conditions compared with marine transport and have relatively fixed schedules. Utilizing these two modes of transport, actual transportation time can be approximated and replaced by theoretical transportation time. However, in the case of opting for marine transport, delays might often occur, so the actual transportation time should be equal to the theoretical transportation time plus the delay time. Here, the delay time for marine transport is approximately represented by the expected delay time.

The transport time of a container from node $i$ to adjacent node $j$ is usually obtained by dividing the transport distance $d_{ij}^{n}$ by the average speed $v^{n}$ . Here, we consider Equation 1 to solve for:

\partial_{ij}^{n} = \frac{d_{ij}^{n}}{v^{n}}, \forall i, j \in N, \forall n \in (W \cup R \cup H)

(1)

The multimodal container transport system is vulnerable to waterborne port congestion. This is frequently caused by adverse weather conditions, leading to unpredictable ship arrival times and frequent delays in marine transport. As a result, delays are challenging to manage effectively ( 27 ).

Assuming that the probability distribution of a ship entry delay follows a negative exponential distribution ( 51 ), the probability density of a ship entry delay with a delay time value of $t$ can be expressed as follows:

f_{ij}^{n} = λ_{ij} \cdot e^{- γ_{ij} t}, \forall i, j \in N, \forall n \in W

(2)

Equation 2 represents the fundamental form of the negative exponential distribution, where $f_{ij}^{n}$ represents the probability density of ship delays between node $i$ and node $j$ , and $λ_{ij}$ and $γ_{ij}$ represent the parameter of the probability density function of ship delays between node $i$ and node $j$ .

The probability of late arrival can be found by calculating the ratio of late arrivals to the number of all arriving ships, which can be expressed as shown in Equation 3.

g_{ij} = \frac{ψ_{ij}}{ω_{ij}}, \forall i, j \in N, \forall n \in W

(3)

where $g_{ij}$ is the probability of a ship being delayed between node $i$ and adjacent node $j$ ; and $ψ_{ij}$ and $ω_{ij}$ represent the number of late ships and the total number of ships sailing from node $i$ to node $j$ , respectively. We use the probability formula:

1 = g_{ij}^{0} + lim_{x \to 0} \int_{t = x}^{t = \infty} f_{ij}^{n} dt, \forall i, j \in N, \forall n \in W

(4)

In Equation 4, the probability of a ship being late, $\lim_{x \to 0} \int_{t = x}^{t = \infty} f_{i j}^{n} d t$ , can be obtained by mathematically integrating the probability density of the ship being late, $f_{ij}^{n}$ . The sum of the probabilities of ship delays and no delays is equal to one. $g_{0}$ is the probability when $t = 0$ , and its value is

g_{ij}^{0} = 1 - g_{ij}, \forall i, j \in N

(5)

Therefore, Equation 4 can also be written as

1 = 1 - g_{ij} + lim_{x \to 0} \int_{t = x}^{t = \infty} f_{ij}^{n} dt, \forall i, j \in N, \forall n \in W

(6)

\begin{matrix} g_{ij} = lim_{x \to 0} \int_{t = x}^{t = \infty} f_{ij}^{n} dt \\ = λ_{ij} \int_{t = x}^{t = \infty} e^{- γ_{ij} t} dt \\ = {- \frac{λ_{ij}}{γ_{ij}} e^{- γ_{ij} t} |}_{t = x}^{t = \infty} \\ = \frac{λ_{ij}}{γ_{ij}} \forall i, j \in N \end{matrix}

(7)

Therefore, $λ_{ij} = γ_{ij} g_{ij}$ , and the delay probability density is

f_{ij}^{n} = γ_{ij} g_{ij} e^{- γ_{ij} t}, \forall i, j \in N, \forall n \in W

(8)

The expected delay time between node $i$ and node $j$ can be expressed as

\begin{matrix} E_{ij}^{n} = \int_{0}^{+ \infty} t \times f_{ij}^{n} dt, \\ = \int_{0}^{+ \infty} t λ_{ij} e^{- γ ijt} dt \\ = \int_{0}^{+ \infty} t γ_{ii} g_{ij} e^{- γ ijt} dt \\ = γ_{ii} g_{ij} \int_{0}^{+ \infty} e^{- γ ijt} dt \forall i, j \in N, \forall n \in W \end{matrix}

(9)

In Equation 9, by knowing any two parameters among $λ_{ij}$ , $γ_{ij}$ , and $g_{ij}$ , it is possible to solve for the expected time of marine delays. Therefore, more practically, the transportation time from node $i$ to adjacent node $j$ for various modes of transport between two nodes can be expressed as

T_{ij}^{n} {\begin{matrix} = \partial_{ij}^{n} x_{ij}^{n} & \forall i, j \in N, \forall n \in (R \cup H) \\ \approx (\partial_{ij}^{n} + E_{ij}^{n}) x_{ij}^{n} & , \forall i, j \in N, \forall n \in W \end{matrix}

(10)

where $\partial_{ij}^{n} x_{ij}^{n}$ represents the theoretical transport time when choosing to transport at node $i$ and adjacent node $j$ , and $E_{ij}^{n} x_{ij}^{n}$ is the delay time when using marine transport.

2. Fixed transit time

In a multimodal transport network, the transit time is assumed to be zero, when the same transport method is used, at a transit node. Transit time is only generated when the transit operation is performed by different transport methods ( 52 ). The fixed transit time at node $j$ is $t_{j}^{nl}$ .

3. Transit storage time

In the international multimodal transport process, rail and road transport are typically scheduled based on fixed timetables. Consequently, adherence to the time window restrictions at transshipment nodes is essential for the arrival of goods subject to this scheduling nature. This ensures avoidance of cost arising from improper connections between the two transportation modes. Generally, nodes allow a certain amount of free storage time, and exceeding this period results in detention storage charges. In such cases, a detailed analysis of the node transition process becomes necessary. If a container arrives at the node before the departure time of the transportation mode, no waiting time is incurred. However, if the container arrives later than the departure time, it must await the next departure schedule. As illustrated in Figure 2, transporting within the time window incurs free transit storage time, while transporting outside the time window results in chargeable transit storage time, as depicted in Figure 3.

Figure 2.

Transit process occurs within the time window.

Figure 3.

Transit process occurs outside the time window.

When a container arrives at a node, an interim transfer operation is initiated. If the container transfer can be completed before the latest departure time of the transfer vehicle’s shift, then no container storage fee will be charged. This means that if the container’s arrival time plus the reloading time and free storage time fall between the earliest and latest departure times of the reloading vehicle’s shift, there is no need to consider container storage time cost. The cargo should arrive at the transit node ahead of time to allow for container transfer, and this corresponds to the allowed time window for arrival. In Figure 2, the blue block represents the range of time allowed for arrival from node $j$ to node $m$ by the transport $l$ . In other words, this is the time window during which storage time and fees are not incurred.

When the container transfer operation at a node is completed, if it cannot meet the departure time of the subsequent transfer vehicle, storage time and costs are incurred. On the arrival of containers at node $j$ , even without free storage, they proceed directly to the transshipment station of the next mode of transport for fixed transfer operations. However, this still does not align with the nearest arrival time window, resulting in charged storage. The yellow blocks in Figure 3 represent the time range allowed for transportation from node $j$ to node $m$ . In other words, it represents the time window during which no storage time and fees are incurred. At this point, the container’s storage time can be expressed using the $t_{j}^{c}$ between the two yellow time windows, as shown in Equations 11 and 12:

t_{j}^{c} = φ_{jm}^{l, a} - (A_{j} + t_{j}^{t} + t_{j}^{f}), ξ_{jm}^{l, a - 1} \leq A_{j} + t_{j}^{t} + t_{j}^{f} \leq φ_{jm}^{l, a}

(11)

When the container arrives at node $j$ , the time $A_{j}$ plus the fixed transit time $t_{j}^{t}$ , the free storage time $t_{j}^{f}$ , has already exceeded the departure time $ξ_{jm}^{l, a - 1}$ of the ( $a$ – $1$ )th train from node $j$ to subsequent node $m$ by transportation mode $l$ . However, it can catch the earliest departure time $φ_{jm}^{l, a}$ of the $a$ th train.

The storage time at this point is calculated as: the time $φ_{jm}^{l, a}$ , the earliest $a$ th train departing from node $j$ to subsequent node $m$ minus the arrival time $A_{j}$ at node $j$ , fixed transit time $t_{j}^{t}$ , and free storage time $t_{j}^{f}$ .

A_{j} = {\begin{matrix} T_{o} j = o, \forall i, j \in N, \\ \forall n \in (W \cup R \cup H) \\ L_{ij}^{n, b} + T_{ij}^{n} others, \forall i, j \in N, \\ \forall n \in (W \cup R \cup H) \end{matrix}

(12)

In Equation 12, when node $j$ is the origin station, the arrival time $A_{j}$ is equal to the departure time $T_{o}$ . However, when this node is an intermediate node, the arrival time $A_{j}$ at node $j$ is equal to the departure time $L_{ij}^{n, b}$ from the previous node $i$ plus the transportation time $T_{ij}^{n}$ from $i$ to $j$ .

4. Total transportation time

The above analysis indicates that the total duration of intermodal transportation from the departure station to the destination station includes the transportation time from node $i$ to adjacent node $j$ , fixed transfer time, and transfer storage time. The mathematical expression is as follows:

\begin{matrix} T_{od} = \sum_{i, j \in N} \sum_{n \in (W \cup R \cup H)} T_{ij}^{n} \\ + \sum_{j \in N \ {o, d}} \sum_{n, l \in (W \cup R \cup H)} y_{j}^{nl} t_{j}^{nl} \\ + \sum_{j \in N \ {o, d}} \sum_{n, l \in (W \cup R \cup H)} y_{j}^{nl} t_{j}^{c} \end{matrix}

(13)

In Equation 13, $\sum_{i, j \in N} \sum_{n \in (W \cup R \cup H)} T_{ij}^{n}$ represents the time spent on transport between adjacent nodes. $\sum_{j \in N \ {o, d}} \sum_{n, l \in (W \cup R \cup H)} y_{j}^{nl} t_{j}^{nl}$ represents the total fixed transfer time of two different transportation modes at intermediate nodes. $\sum_{j \in N \ {o, d}} \sum_{n, l \in (W \cup R \cup H)} y_{j}^{nl} t_{j}^{c}$ represents all storage time.

While the expected time delay from using marine transport is considered in the overall transport process, this time does not truly represent the actual duration of marine transport, but serves as an approximate substitute. After the completion of marine transport, the range of impacts from delays cannot be effectively and accurately estimated. The expected delay time, as a substitute, can only serve as a basic reference. Although a feasible transportation route has been identified, the timely connection of different transportation modes at subsequent nodes remains unknown. Therefore, there is uncertainty in the storage time based on delays. Even for the same transportation route, significant differences in storage time at nodes exist because of variations in the magnitude of delays in marine transport. Given the substantial impact of the occurrence of delays on subsequent route selection, this study treats the total transportation time as an uncertain variable, and represents it effectively using an uncertainty function.

In the decision-making process, uncertain variables can be described using uncertain distributions. The uncertain distribution $Φ$ of an uncertain variable $η$ is defined as $Φ (x) = {η \leq x}$ where $x$ is any real number ( 38 ). The uncertainty measure differs essentially from the probability measure, and the connotation of the uncertain variable is also different from that of the random variable. In fact, uncertainty measurement can be understood as the reliability of the occurrence of an uncertain event, or the chance of an imprecise objective event, rather than the probability. It depends on the degree of knowledge of the event by experts and has a certain subjectivity ( 38 , 40 ). To better comprehend the uncertainty theory referenced in this paper, the Online Appendix introduces relevant theorems, definitions, and corollaries. Under a certain confidence level, $α_{od}$ , the total transportation time of the identified container shipping route that meets the actual transport time requirements can be expressed as:

M {T_{od} \leq η_{od}} \geq α_{od}, \forall o, d \in N

(14)

Deterministic Equivalence Class Model

The actual total transportation time, $η_{od}$ , for intermodal container transportation from the origin to the destination is an uncertain variable with a continuous distribution function. Its uncertain distribution function is denoted as $Φ_{η_{od}} (x)$ , $\forall o, d \in N$ ( 46 ). The equivalence class of the relevant opportunity constraints can be determined based on the definitions, theorems, and corollaries mentioned in the Online Appendix ( 46 ).

M {T_{od} \leq η_{od}} = 1 - M {T_{od} \geq η_{od}} \geq α_{od}, \forall o, d \in N

(15)

when, and only when,

T_{od} \leq f_{η_{od}} (α_{od}), \forall o, d \in N

(16)

where $η_{od}$ is the inverse uncertain distribution of the uncertain variable $f_{η_{od}} (α_{od}) = Φ_{η_{od}}^{- 1} (α_{od}) • (1 - α_{od})$ , $\forall o, d \in N$ , as defined by Liu and Ha ( 46 ). Therefore, Equation 16 can also be written as Equation 17. Let us assume that the total multimodal transportation time $η_{od}$ from origin to destination is a zigzag uncertain variable, denoted as $Z (a_{od}, b_{od}, c_{od})$ . According to Inference 1, we can obtain the inverse uncertainty distribution of $η_{od}$ as Equation 18.

T_{od} \leq Φ_{η_{od}}^{- 1} (α_{od}) • (1 - α_{od}), \forall o, d \in N

(17)

\begin{matrix} Φ_{η_{od}}^{- 1} (α_{od}) = {\begin{matrix} (1 - 2 α_{od}) \cdot a_{od} + 2 \cdot α_{od} \cdot b_{od}, \\ if α_{od} < 0.5 \\ (2 - 2 α_{od}) \cdot b_{od} + (2 α_{od} - 1) \cdot c_{od}, \\ if α_{od} \geq 0.5 \end{matrix} \end{matrix}

(18)

Related Cost for Multimodal Container Transport

The cost of international intermodal container transportation consists of three main components: transportation cost from node $i$ to adjacent node $j$ , fixed transit cost, and storage cost generated at the transit node. For international freight forwarders, in addition to the total cost of transportation, there will also be corresponding costs of customs operations and personnel, but the differences in such costs generated by different routes is small. More importantly, it does not affect the specific route selection. Therefore, we do not consider this part of the cost in this paper.

1. Transportation cost from node $i$ to adjacent node $j$

International intermodal container transportation employs various modes, including marine, road, and rail transport, and each mode has different freight rate costs. For rail and road container transport, the freight calculation is similar, involving multiplying the freight rate by the billable quantity of the containers, $q$ , and adding transportation surcharges, $G_{ij}^{n}$ . The freight rate consists of two components: basic freight rate 1, $s_{ij}^{n 1}$ , and basic freight rate 2, $s_{ij}^{n 2}$ . Basic freight rate 1, $s_{ij}^{n 1}$ , represents the freight rate per unit of container number when sending and receiving goods at the station, while basic freight rate 2, $s_{ij}^{n 2}$ , represents the freight rate per unit of box number per freight kilometer during the transit period. For railway transport, there are container surcharges $G_{ij}^{n} (\forall n \in R)$ including electrified railway, railway construction funds, and specific freight charges. Road transportation fees $G_{ij}^{n} (\forall n \in H)$ are determined based on the specific conditions of container cargo transportation, and carriers may charge additional fees such as for shunting, delay, loss for lost containers, road blocking parking, vehicle disposal, and other surcharges. The transportation cost for rail or road transport between two nodes can be expressed using Equation 19.

\begin{matrix} S_{ij}^{n} = s_{ij}^{n 1} \times q + s_{ij}^{n 2} \times d_{ij}^{n} \times q \\ + \sum G_{ij}^{n}, \forall i, j \in N, \forall n \in (R \cup H) \end{matrix}

(19)

where $s_{ij}^{n 1} \times q$ constitutes the total basic rate 1, and $s_{ij}^{n 2} \times d_{ij}^{n} \times q$ constitutes the total basic rate 2. $\sum G_{ij}^{n} (\forall n \in (R \cup H))$ represents the total composition of rail (or road) surcharges.

The cost of container marine transport also includes basic freight rates and various surcharges. Its structure is similar to that of traditional breakbulk liner freight. Unlike rail transport and road transport, most container liner companies now use the package rate system for basic rates. In addition, there are various additional costs incurred during transportation. The cost of using marine transport between two adjacent nodes can be expressed by Equation 20.

S_{ij}^{n} = s_{ij}^{n 3} \times q + \sum G_{ij}^{n}, \forall i, j \in N, \forall n \in W

(20)

where $s_{ij}^{n 3} \times q$ constitutes the total basic fee for marine transport, while $\sum_{n \in W} G_{ij}^{n}$ represents the total composition of the marine transport surcharge.

2. Fixed transit cost

The total fixed transit cost at node $j$ is determined by the unit fixed transit cost and the volume of goods transported. Equation 21 can be used to calculate its replacement fee at the transfer station.

δ_{j} = C_{ij}^{nl} p, \forall i, j \in N, \forall n, l \in (W \cup R \cup H)

(21)

where $C_{j}^{nl}$ is the unit transport cost of converting transport mode $n$ to $l$ at node $j$ , and $p$ is the volume of goods transported.

3. Storage cost

From the transit storage time analyzed above, it can be inferred that storing containers at transit nodes incurs additional storage costs, based on the storage time. Demurrage storage cost arises after a container has been transported to point $j$ through the previous mode of transport, and any free storage period expires before it is possible to catch the most suitable shift of the transportation mode to the next node. It is necessary to wait for a certain period of time at point $j$ . This period of time requires the payment of the corresponding storage fee. For rail and road transport, container storage at transit nodes takes place at the respective railway yard or road freight depot, while in marine transport storage takes place at the port. Equation 22 gives the calculation of the total storage cost at a node:

Q_{j} = {pt}_{j}^{c} ρ_{j}, \forall j \in N

(22)

where $t_{j}^{c}$ represents the storage time fee for the previously analyzed node $j$ , and $ρ_{j}$ is the unit storage fee generated at node $j$ .

4. Total transportation cost

The transportation cost corresponds to the preceding transportation time component. This includes the transportation cost from node $i$ to adjacent node $j$ , fixed transit cost, and transit storage cost. This can be mathematically expressed as follows:

\begin{matrix} C_{od} = \sum_{i, j \in N} \sum_{n \in (W \cup R \cup H)} x_{ij}^{n} S_{ij}^{n} p \\ + \sum_{j \in N \ {o, d}} \sum_{n, l \in (W \cup R \cup H)} y_{j}^{nl} δ_{j} \\ + \sum_{j \in N \ {o, d}} \sum_{n, l \in (W \cup R \cup H)} y_{j}^{nl} Q_{j} \end{matrix}

(23)

In Equation 23, $\sum_{i, j \in N} \sum_{n \in (W \cup R \cup H)} x_{ij}^{n} S_{ij}^{n} p$ is the transportation cost between all adjacent nodes in a transport route. $\sum_{j \in N \ {o, d}} \sum_{n, l \in (W \cup R \cup H)} y_{j}^{nl} δ_{j}$ is the fixed transit cost of all nodes. $\sum_{j \in N \ {o, d}} \sum_{n, l \in (W \cup R \cup H)} y_{j}^{nl} Q_{j}$ is the total storage cost.

Model Formulation

Based on the above analysis and considering marine transport delay time, we have developed a route optimization model for multimodal container transport.

Objective functions:

Min Z_{1} = T_{od}

(24)

Min Z_{2} = C_{od}

(25)

Objective function $Z_{1}$ is to minimize the total transportation time and objective function $Z_{2}$ is to minimize the transportation cost, subject to Equations 1, 9–13, 17–23 and the following additional constraints:

The multimodal transport must have a complete route from the origin to the destination.

\sum_{\binom{i, j \in N}{i \neq j}} x_{ij}^{n} - \sum_{\binom{i, j \in N}{i \neq j}} x_{ji}^{n} = {\begin{matrix} 1, & i = o \\ - 1, & i = d \forall n \in (W \cup R \cup H) \\ 0, & others \end{matrix}

(26)

Equation 26 ensures that the transport of goods begins at the originating station and ends at the terminating station, with no interruption in the process of transport at intermediate nodes. Furthermore, only one transportation mode can be used between any nodes:

\sum_{n \in (W \cup R \cup H)} x_{ij}^{n} \leq 1, \forall i, j \in N, i \neq j

(27)

Equation 27 indicates transportation mode choice constraints, where $\sum_{n \in (W \cup R \cup H)} x_{ij}^{n}$ represents the summation of all transportation modes at the same node.

The number of transfers allowed per node is limited to one. Equation 28 represents this transit constraint:

\sum_{nl \in W \cup R \cup H} y_{j}^{nl} \leq 1, \forall i, j \in N

(28)

During the transportation of goods, the continuity of the transportation process should be guaranteed. Equation 29 is a continuity constraint on the transport process:

\begin{matrix} x_{ij}^{n} + x_{jm}^{l} \geq 2 y_{j}^{nl}, \forall i, j, m \in N, \forall n, l \in (W \cup R \cup H) \end{matrix}

(29)

Taking into account that shippers of different goods have different time and cost requirements, they have different thresholds for the total transportation time and cost that they can bear as a result of delays.

T_{od} \leq T_{od}^{\max}, \forall o, d \in N

(30)

C_{od} \leq C_{od}^{\max}, \forall o, d \in N

(31)

where $T_{od}^{\max}$ and $C_{od}^{\max}$ are the maximum total transportation time and cost that the shipper can bear, respectively. When shippers have only time requirements with no cost constraints, the $C_{od}^{\max}$ in Equation 31 can take a value of positive infinity. Similarly, when shippers prioritize cost with no time constraints, the variable $T_{od}^{\max}$ in Equation 30 can take a value of positive infinity. If there are requirements for both time and cost, the values depend on the specific circumstances.

The value constraints of the decision variables are as follows:

\begin{matrix} \partial_{ij}^{n}, f_{ij}^{n}, T_{ij}^{n}, T_{od}, C_{od}, η_{od}, S_{ij}^{n 1}, S_{ij}^{n 2}, S_{ij}^{n 3}, G_{ij}^{n}, C_{j}^{nl}, δ_{j}, \\ T_{od}^{\max}, C_{od}^{\max}, a_{od}, b_{od}, c_{od} \in (0, + \infty) \end{matrix}

(32)

E_{ij}^{n}, t_{j}^{c}, t_{j}^{t}, t_{j}^{f}, Q_{j} \in [0, + \infty)

(33)

g_{ij}^{0}, g_{ij}, α_{od} \in [0, 1]

(34)

x_{ij}^{n}, y_{j}^{nl} \in {0, 1}, \forall i, j \in N, \forall n, l \in (W \cup R \cup H)

(35)

ADE Algorithms Incorporating Learning Mechanisms

The multimodal transportation route optimization problem is a typical non-deterministic polynomial-time hard (known as “NP-hard”) problem, known for its high computational complexity ( 53 ). As the problem size increases, the solution time and storage requirements exhibit an exponential growth trend, rendering the use of exact solving methods infeasible. Therefore, this paper attempts to employ an intelligent algorithm for solving the problem. The differential evolution algorithm has gained wide applicability, because of its simplicity, fast convergence, and robustness (54 –57). However, the differential evolution algorithm also faces two main challenges. First, the choice of different crossover and mutation strategies can have a significant impact on the results, and determining the optimal strategy can be time-consuming. Second, the population is prone to premature convergence, becoming trapped in local optima.

To tackle these issues, this study introduces a novel ADE algorithm with an incorporated learning mechanism. The algorithm dynamically adjusts crossover and mutation operators using a sliding window approach, enabling continuous learning and selection of optimal combination strategies. Furthermore, a local search strategy is implemented to explore and escape local optima, thereby enhancing the algorithm’s global search capability. The specific steps of the algorithm designed in this study are outlined as follows:

Step 1: Encoding and decoding of evolved individuals

Before optimization, the problem space needs to be mapped to the solution space. The solutions of the problem are first encoded as chromosomes, with each encoding corresponding to a solution ( 55 ).

Taking container intermodal transport from Japan to Europe in five stages as an example, we explain the detailed operational steps of the algorithm. The first four digits present the transportation nodes, and the first digit is the serial number of the Japanese port. For Japan Port 1, the code of transportation node $l$ is 1. The remaining three digits represent Chinese port, departure cities, and arrival cities respectively. The encoding method is the same as the first digit. The last five digits represent transport methods in stages 1–5, respectively: 1 means rail, 2 means road, and 3 means marine. The chromosome coding composition is illustrated in Figure 4.

Figure 4.

Encoding method.

Step 2: Generation of the initial total cluster

The quality of the initial solution greatly affects the solution speed and convergence quality of the algorithm. Therefore, when the initial population is generated, if the quality of the initial solution can be improved under the premise of ensuring the randomness of the population, the efficiency of the algorithm will be effectively improved. Therefore, in this study, roulette wheel selection was employed to generate the initial population, taking into account the similarity between individual genes. This approach aims to enhance the quality of the initial population.

Step 3: Calculation of fitness values

The model in this paper is a dual-objective nonlinear integer programming model that aims to minimize both transportation time and cost. To address differences in units and orders of magnitude between the two objectives, we employ the ideal point method to transform the multi-objective problem into a single objective. This ideal point method calculates the distance between the feasible solution set and the ideal solution set. It then selects the solution in the feasible set with the closest distance to the ideal solution, determining the optimal solution ( 57 ). This method is effective in solving multi-objective decision-making problems. First, we focus on a single objective, disregarding other objectives, and obtain its optimal solution under identical constraint conditions. Through this step-by-step solving approach, we derive the individual components of the optimal solution $Z^{min}$ .

Z^{min} = (z_{1}^{min}, \dots, z_{m}^{min})

(36)

According to the shortest distance ideal point method, the evaluation function can be defined as follows ( 57 ):

U (z_{1}, \dots, z_{m}) = \sqrt{\sum_{j = 1}^{m} {(z_{j} - z_{j}^{min})}^{2}}

(37)

Since the total transportation cost is much greater than the total transportation time in this paper, the evaluation function mentioned above is modified as follows:

U^{'} (z_{1}, \dots, z_{m}) = \sqrt{\sum_{j = 1}^{m} {(\frac{z_{j} - z_{j}^{min}}{z_{j}^{min}})}^{2}}

(38)

We then transform the original multi-objective optimization problem into the following single-objective optimization problem:

Z = min U^{'} (z_{1}, \dots, z_{m})

(39)

The fitness values of each chromosome can be obtained by calculating according to the above formula.

Step 4: Mutation and crossover

The mutation operation in the differential evolution algorithm is to randomly select evolving individuals $a$ , $b$ for differential operation, then, after scaling by a certain integer multiple of $f$ , it is combined with the target individual $x$ to form a mutated individual $v$ . The calculation formula is as follows:

Chrom (v) = Chrom (x) + f \times [Chrom (a) - Chrom (b)]

(40)

For example, if we set $f = 0.3$ , $Chrom (x) = [3, 4, 2, 4, 2, 3, 2, 1, 2]$ , $Chrom (a) = [3, 3, 2, 4, 2, 3, 1, 1, 2]$ , $Chrom (b) = [3, 2, 5, 4, 1, 3, 2, 1, 1]$ , the mutation process shown in Figure 5 can be obtained.

Figure 5.

The mutation process.

In the mutation operation, the target individual $x$ is crossed with the mutant individual $v$ , with a certain probability for certain gene segments, resulting in a new individual $u$ . To improve the efficiency of the algorithm, this study introduces the adaptive crossover probability, that is, as the population evolves iteratively, the crossover rate also changes. The adaptive crossover probability (CP) can be calculated by Equation 41.

CP = C P_{min} + cg \frac{C P_{max} - C P_{min}}{G},

(41)

where $C P_{min}, C P_{max}$ represent the minimum and maximum cross probability, $cg$ represents the current evolutionary generation, and $G$ represents the total evolutionary generations. The cross process is shown in Figure 6.

Figure 6.

The cross process.

Step 5: Learning-based adaptive crossover and mutation combination strategy selection mechanism.

Within time $t$ , record the count $H$ and corresponding reward values $Q_{t} (i)$ of all adaptive crossover and mutation combination strategies. In the initial stage, the fitness values are used as the reward values for the combination strategies, $Q_{t} (i) = Z$ . Next, calculate the average reward, $\bar{Q_{t} (i)}$ , for all combination strategies, identify the count $m_{t}$ of crossover and mutation strategies with rewards below this average. The exploration probability $p$ is calculated by utilizing a parameter $τ$ , which signifies the decay rate for the exploration probability.

\bar{Q_{t} (i)} = \frac{\sum_{i = 1}^{H} Q_{t} (i)}{H}

(42)

p = \frac{τ}{τ + m_{t}^{2}}

(43)

To make a selection, generate a random number $rand$ within the open interval (0,1) and compare it with $p$ . If $rand$ is greater than $p$ , select the crossover and mutation combination strategy with the highest average reward value $A_{t} = \arg_{i} max Q_{t} (i)$ . If $rand$ is less than or equal to $p$ , randomly choose a crossover and mutation combination strategy $A_{t} = \arg_{i} rand {Q_{t} (i)}$ . Calculate the difference $f$ between the average fitness values before and after. After applying the selected crossover and mutation combination strategy, use $f$ as the reward value.

Q_{t} (i) = f = | Z^{after} - Z^{before} |

(44)

Step 6: Combined policy update approach.

Check if the number of iterations is less than or equal to the length $(L)$ of the learning combination strategy selection record table. If it is, store the selected cross-variation combination strategy and its corresponding reward value in the table. Otherwise, remove the first row from the strategy selection record table, shift the remaining rows up, and log new data in the last row.

Step 7: Local search operation.

Perform a local search on a randomly selected subset of $⌊ N_{p} \times vn \times {(\frac{cg}{100})}^{2} ⌋$ individuals from the record table. Here, $vn$ represents the variable neighborhood search parameters, and $cg$ represents the current iteration number. The variable neighborhood operation is applied to the chosen individuals, incorporating the following strategies:

Local search operation. Perform a local search on a randomly selected subset of $m$ individuals from the record table. Here, $m_{1}$ represents the variable neighborhood search parameters, $m_{2}$ represents the current iteration number.

Two-opt strategy. Choose a chromosome randomly and select two nodes or two different transport modes, then swap the coding numbers of the selected nodes or modes.

Random variable negation strategy. Choose a chromosome randomly and select a transport node or mode, then replace the encoded number with a different one.

Internal exchange neighborhood structure. Randomly choose a chromosome and select multiple gene segments with similar characteristics that are interconnected, then perform an exchange operation on these segments.

External insertion neighborhood structure strategy. Randomly select a chromosome and choose multiple gene segments with similar properties that are interconnected, then remove these segments and insert an equivalent number of new gene segments.

Step 8: Solution repair.

After completing the local operation, assess the feasibility of each solution. If there is a solution that violates constraints or does not meet real-world requirements, then repeat the variable neighborhood operation mentioned above until all solutions become feasible. Check if the number of local search individuals is less than $⌊ N_{p} \times vn \times {(\frac{cg}{100})}^{2} ⌋$ . If it is, perform the local search operation. Otherwise, return to Step 3.

Case Analysis

Data

To validate the effectiveness of the designed model and algorithm, we chose container multimodal transport from Osaka (Japan) to Minsk (Belarus) as a case study. Because of geographical constraints or station limitations, the selection of transportation modes is restricted in certain segments. In particular, inland container transportation primarily occurs in China and arrival cities, while waterborne port container transportation, in addition to this, also includes Japanese cities. The unit transportation costs for inland and maritime port containers are illustrated in Figure 7, a and b , respectively. The multimodal transportation network is depicted in Figure 8. The average transit times for transit nodes in different cities are presented in Table 3. Parameters for the probability density of marine transport delays between different ports are shown in Tables 4 to 6. Table 7 provides the uncertain total transportation time for multimodal transport from Japanese to European arrival cities. We analyzed actual operational data from Osaka Port to Minsk Port for 30 days, covering the periods from March 1 to March 10, 2021, April 1 to April 10, 2021, and May 1 to May 10, 2021.

Figure 7.

The unit transit cost for inland transport and waterborne port containers: (a) inland containers and (b) waterborne port containers.

Figure 8.

Transportation route from Osaka to Minsk.

Table 3.

Average Transit Time of Transit Nodes

Transit nodes	Transit method (hours)
Transit nodes	Rail to sea	Road to sea	Sea to road	Sea to rail	Road to rail	Rail to road
Japanese ports	17.33	15.63	NA	NA	NA	NA
Chinese ports 1	NA	NA	13.88	12.73	NA	NA
Chinese ports 2	17.56	10.25	11.69	16.21	10.22	9.73
Arrival cities	NA	NA	16.23	14.71	9.88	10.76

Note: NA = Not available.

Table 4.

Parameter Values $(λ_{ij}, γ_{ij})$ of the Delay Probability Density of Marine Transport Between Japanese Ports and Chinese Ports

Ports	Shanghai	Ningbo	Tianjin	Nanjing	Lianyungang	Qingdao
Nagoya	(0.009,0.019)	(0.010,0.025)	(0.010,0.025)	(0.011,0.018)	(0.015,0.026)	(0.018,0.018)
Kobe	(0.013,0.020)	(0.012,0.026)	(0.015,0.034)	(0.009,0.018)	(0.007,0.017)	(0.012,0.021)
Osaka 2	(0.015,0.025)	(0.019,0.028)	(0.012,0.024)	(0.012,0.031)	(0.013,0.023)	(0.014,0.019)
Yokohama	(0.009,0.018)	(0.011,0.025)	(0.012,0.031)	(0.010,0.022)	(0.011,0.034)	(0.011,0.023)
Tokyo	(0.011,0.023)	(0.009,0.025)	(0.010,0.028)	(0.014,0.024)	(0.013,0.028)	(0.013,0.022)

Table 5.

Parameter values $(λ_{ij}, γ_{ij})$ of the Delay Probability Density of Marine Transport Between Chinese Ports

Ports	Changsha	Changchun	Wuhan	Chongqing	Chengdu
Shanghai	(0.006,0.014)	(0.007,0.017)	(0.005,0.015)	(0.007,0.019)	(0.012,0.015)
Ningbo	(0.008,0.015)	(0.005,0.018)	(0.008,0.017)	(0.005,0.018)	(0.010,0.017)
Tianjin	(0.005,0.019)	(0.006,0.015)	(0.007,0.021)	(0.011,0.016)	(0.012,0.016)
Nanjing	(0.008,0.015)	(0.006,0.013)	(0.006,0.013)	(0.009,0.022)	(0.008,0.020)
Lianyungang	(0.006,0.014)	(0.007,0.015)	(0.007,0.014)	(0.005,0.014)	(0.007,0.013)
Qingdao	(0.007,0.013)	(0.009,0.014)	(0.008,0.016)	(0.006,0.015)	(0.005,0.014)

Table 6.

Parameter Values $(λ_{ij}, γ_{ij})$ of the Delay Probability Density of Marine Transport Between Chinese Ports and European Ports

Ports	Rhodes	Warsaw	Hamburg Port	Minsk 1	Moscow	Malaszewicze
Changsha	(0.023,0.035)	(0.021,0.034)	(0.024,0.029)	(0.025,0.028)	(0.028,0.035)	(0.025,0.036)
Changchun	(0.025,0.034)	(0.022,0.032)	(0.027,0.035)	(0.026,0.029)	(0.027,0.034)	(0.027,0.033)
Wuhan	(0.027,0.035)	(0.028,0.035)	(0.031,0.028)	(0.028,0.034)	(0.025,0.032)	(0.029,0.031)
Chongqing	(0.028,0.038)	(0.024,0.030)	(0.029,0.029)	(0.031,0.038)	(0.023,0.028)	(0.028,0.029)
Chengdu	(0.024,0.037)	(0.022,0.033)	(0.022,0.034)	(0.026,0.037)	(0.024,0.033)	(0.025,0.037)

Table 7.

Uncertain Total Transportation Times for Multimodal Transport from Japanese to European Final Destination Cities (hours)

Ports	Rhodes	Warsaw	Hamburg Port	Minsk 1	Moscow	Malaszewicze
Nagoya	(495,645,744)	(498,636,762)	(495,656,753)	(501,630,725)	(493,623,731)	(489,646,742)
Kobe	(486,651,742)	(498,644,757)	(486,648,751)	(499,646,741)	(485,660,756)	(494,643,762)
Osaka 2	(492,642,753)	(502,652,762)	(498,656,754)	(502,648,753)	(488,756,748)	(493,658,751)
Yokohama	(507,648,757)	(500,652,754)	(481,644,742)	(515,626,766)	(497,632,757)	(498,640,756)
Tokyo	(487,630,762)	(483,642,754)	(496,652,756)	(501,641,747)	(496,632,759)	(490,646,750)

Calculation Process and Result Analysis

Based on the preceding description, a differential evolution algorithm was employed to address the problem, with the coding rules detailed in Figure 9.

Figure 9.

Multimodal transport coding rules for container trains between Japan and Europe.

Based on 30 initial populations, with a CP of 0.9 and a variation probability of 0.08, we set the maximum number of iterations to 600. For the sake of this analysis, we will initially disregard the shippers’ tolerance thresholds for delay time and cost. Subsequent analysis will specifically address the critical thresholds for time and cost. To increase the confidence level of the obtained total transportation time, this paper first sets the uncertainty confidence level $α_{od}$ to be 0.95. The evolution process in Figure 10 shows that the algorithm stabilizes after less than 400 iterations, indicating that the algorithm has reached convergence and the optimal route is obtained.

Figure 10.

Convergence process.

To illustrate better the superiority of the proposed model, first, we compared it with the actual transportation routes and a deterministic model that disregards transportation delays. The deterministic model can be obtained by eliminating equations related to marine transport delays. The transportation routes for three scenarios are presented in Table 8, while the corresponding comparisons of total transportation time and cost for three months are depicted in Figure 11. The broken lines in Figure 11 represent total transportation time, while the bars depict total transportation cost. From Figure 11 it is evident that both the deterministic model and the proposed model effectively reduce transportation time and cost. The deterministic model performs best in reducing transportation time.

Table 8.

Route Optimization Results in Different Months

Date	Route category	Route code
Date	Route category	March	April	May
1	Actual	115623312	516213211	326223312
	Deterministic	421323312	414423212	541323131
	Uncertain	325423312	454223111	243623131
2	Actual	536413312	621413212	434423112
	Deterministic	131413112	414423222	415223131
	Uncertain	346523312	355423312	333423312
3	Actual	564223112	136413312	325413331
	Deterministic	323323312	311423231	126413132
	Uncertain	334523112	314323112	655413212
4	Actual	645113331	323413332	244213311
	Deterministic	314213122	261413312	133223311
	Uncertain	323423312	343423312	353523322
5	Actual	352213312	314413312	336323331
	Deterministic	451413212	351413311	322213231
	Uncertain	313523211	352323221	343423312
6	Actual	353423332	564223112	422213331
	Deterministic	541213122	322113332	326123131
	Uncertain	311413312	345413212	326413312
7	Actual	566323111	254423112	311613312
	Deterministic	334323221	325123122	335413222
	Uncertain	353423312	525223312	351323111
8	Actual	313423112	461123111	121413312
	Deterministic	521223121	313113121	445223111
	Uncertain	332223132	546413312	245213231
9	Actual	433413312	354413212	614623111
	Deterministic	533413312	532213231	354223111
	Uncertain	445113212	352423312	432123231
10	Actual	112413112	512313332	414423212
	Deterministic	341123211	541323131	212223112
	Uncertain	335223112	546313122	313423312

Figure 11.

Comparison of deterministic and proposed model solutions with actual transport routes: (a) March, (b) April, and (c) May.

Calculations indicate that, on average, the proposed model reduces transportation time by 56.04 h and the deterministic model by 65.17 h compared with the actual routes. The difference between the two is 9.13 h, which is relatively small. On the other hand, the proposed model excels in reducing transportation cost, averaging a total transportation cost reduction of $511.73 per transportation route compared with the actual routes. In contrast, the deterministic model reduces cost by only $202.57 per route, less than half of the reduction achieved by the proposed model. However, it is essential to note that transportation routes obtained by the deterministic model may not always be feasible. An analysis of the transportation routes for the three scenarios is provided below.

Take the transportation route on March 1 as an example. The actual route is as follows: departure from Osaka 1, transported via road to Nagoya, followed by a road-to-water transshipment in Nagoya. From Nagoya, marine transport proceeds to Shanghai and then continues by water to Chongqing. After transshipment to railway, the container awaits the departure of the China–Europe freight train to Malaszewicze, scheduled for Mondays. However, because of weather and port congestion, the first marine transport segment experienced a delay of 19.11 h, and the second marine transport segment faced an 8.08 h delay. Consequently, the container’s transfer to railway at Chongqing Station concluded at 17:05 on Monday, while the China–Europe freight train from Chongqing to Malaszewicze only departs on Mondays, Thursdays, and Saturdays at 12:49. Consequently, the container had to be stored at Chongqing Station for an additional 67.73 h, waiting for the Thursday departure of the China–Europe freight train. The total transportation time initially planned was 582.36 h. However, because of marine transport delays leading to container storage at the intermodal transfer node, the actual total transportation time amounted to 650.09 h.

If we disregard delays in marine transport, the optimal transportation route obtained from the deterministic model is as follows. Departure from Osaka 1, transported via road to Yokohama; marine transport from Yokohama to Ningbo. Without any transshipment, it then proceeds by water to Changsha. On completion of transshipment at Changsha, it awaits the China–Europe freight train scheduled for 21:20 on the same day, travels by railway to Hamburg, and finally arrives at Minsk 2 via road transport. However, because of the lack of consideration for potential transportation delays, this transportation route exhibits low robustness. Using marine transport, there is a high probability of encountering delays during the transportation process. The expected delay for the first marine transport segment is approximately 18.77 h, and for the second marine transport segment, it is approximately 8.63 h. As a result, the container is likely to arrive at the Changsha railway station on either Friday or Saturday, whereas trains from Changsha to Hamburg only depart on Thursdays, not on other days. The cost and time implications of storing containers at Changsha Station for nearly a week are substantial. As a result, transport routes derived from deterministic modeling may not apply well in real situations, because of possible delays in marine transport.

Compared with the deterministic model, the proposed model exhibits greater robustness by accounting for potential delays in marine transport. The optimal transportation route obtained from the proposed model is as follows. Departure via road transport from the point of dispatch to Osaka, followed by transshipment to marine transport for the journey to Ningbo. Subsequently, without the need for transshipment, it continues by marine transport to Chongqing. At Chongqing Station, it waits for the China–Europe freight train to Minsk, and finally reaches its destination via road transport from Minsk. The proposed model takes into account delays totaling around 25 h for the two waterborne sections. Moreover, the China–Europe freight train from Chongqing to Minsk departs on Mondays, Thursdays, and Saturdays. Therefore, regardless of the extent of delays in marine transport, the storage waiting time for the container will not exceed 48 h. The transportation route derived from the proposed model is superior and more reliable.

Observing Table 8 reveals that the transportation routes for Japan–Europe intermodal containers typically involve initial marine transport from Japan to China, followed by reliance on China–Europe freight train transportation to Europe. This pattern arises because, although marine transport from China to Europe tends to be cost-effective, it lacks assured transportation times. On the other hand, while road transport from China to Europe is faster, it tends to be relatively expensive. Therefore, a combination of marine and rail transport becomes a preferable choice. Consequently, inevitable transitions from water to rail or water–road–rail connections occur in the transportation process. Given the infrequent departure schedules of China–Europe freight trains, typically once a week or every few days, delays in marine transport can generate significant storage time and cost at transshipment nodes. Ultimately, we would emphasize, even for identical transportation routes, the resulting transportation time and cost may vary. For example, there were differences in time and cost for the actual transport routes for March 3 and April 6, and for the transport routes for April 4 and May 5 based on the proposed model solution. These discrepancies are the effects of marine transport delays and rail departure schedules. Therefore, strategically arranging the selection of transportation nodes and modes becomes critical when considering the occurrence of transportation delays.

Shippers have varied expectations for time and cost, depending on the cargo category. For instance, perishable goods demand a tight delivery deadline while cost considerations are lower. Conversely, for items like lumber or steel, shippers prioritize cost over time. Delays in marine transport during multimodal transit introduce significant uncertainty in both total transportation time and cost. To mitigate potential losses, shippers typically set specific requirements for both total transportation time and cost. Therefore, we further explore the threshold value of time and cost, which is what shippers can bear in delays by marine transport, that is, to consider the total transport time and cost that they can accept in case of delay occurrence.

Four new sets of comparative experiments are added to this paper. They are as follows: (i) requiring delivery within a specified time with the cost as small as possible; (ii) requiring minimal transport cost with delivery time as short as possible; (iii) gradually relaxing both delivery time and cost; (iv) relaxing one of delivery time or cost, contracting the other. Using March 1 as an example, the results of the four groups of experiments are shown in Tables 9 to 12.

Table 9.

The Delivery Has a Time Limit and the Cost Should Be as Low as Possible

Constraints	Optimal transportation route	Total transportation time (hours)	Total transportation cost ($)
≤550 h	314423212	544	5,911
≤560 h	323523112	552	5,762
≤570 h	442423112	561	5,632
≤580 h	525413312	572	5,457
≤590 h	345623212	586	5,340
≤600 h	325423312	598	5,212

Table 10.

The Transportation Cost Is Restricted, the Transportation Time Should Be as Short as Possible

Constraints	Optimal transportation route	Total transportation time (hours)	Total transportation cost ($)
≤$5,000	331323312	625	4,932
≤$5,200	455423312	593	5,015
≤$5,400	343423112	578	5,246
≤$5,600	356423112	564	5,538
≤$5,800	323523112	552	5,762
≤$6,000	364423112	541	5,978

Table 11.

Gradual Relaxation of Transportation Time and Cost

Constraints	Optimal transportation route	Total transportation time (hours)	Total transportation cost ($)
≤550 h & ≤$5,000	NA	NA	NA
≤560 h & ≤$5,200	NA	NA	NA
≤570 h & ≤$5,400	215523312	568	5,397
≤580 h & ≤$5,600	356423112	564	5,538
≤590 h & ≤$5,800	611423112	562	5,621
≤600 h & ≤$6,000	611423112	562	5,621

Note: NA = Not available.

Table 12.

Loosen Transportation Time or Cost, Tighten the Other

Constraints	Optimal transportation route	Total transportation time (hours)	Total transportation cost ($)
≤540 h & ≤$6,100	NA	NA	NA
≤550 h & ≤$6,000	543413112	543	5,954
≤560 h & ≤$5,800	466513122	551	5,791
≤570 h & ≤$5,600	356423112	564	5,538
≤580 h & ≤$5,400	343423112	578	5,246
≤590 h & ≤$5,200	355423312	589	5,188
≤600 h & ≤$5,000	NA	NA	NA

Note: NA = Not available.

From Tables 9 and 10, it is evident that as, the total transportation time gradually relaxes, the transportation cost decreases. Conversely, as the total transportation cost gradually relaxes, the total transportation time decreases. Table 11 indicates that, when the expected total transportation time and cost are exceptionally small, the proposed model cannot provide the optimal transportation route. As the total transportation time and cost gradually relax, the model begins to exhibit optimal solutions. However, when the constraints become overly lenient, the optimal solution stabilizes. The results of the final experiment set demonstrate that, when either the total transportation time or cost becomes excessively small, the model becomes infeasible. The optimal transportation time and cost exhibit opposite trends.

Furthermore, to validate the proposed algorithm, we compared target values obtained from ADE algorithm, GA, and the Dijkstra method, as shown in Figure 12. To ensure fairness, the uncertainty level was set at 0.95. The three algorithms exhibit similar trends in transportation time and cost over three months, with the proposed ADE algorithm consistently yielding lower values than GA and Dijkstra. Additionally, we compared average transportation time and cost for three months. Figure 13 illustrates some differences in average transportation cost and time, with the optimal solutions from all three algorithms being best in May. GA had the highest transportation cost in March, while ADE and Dijkstra had the highest in April. Dijkstra and ADE had the longest transportation time in March, and GA had the longest in May. In conclusion, the starting time of container transportation significantly influences total transportation cost and time. To obtain an optimal route, transportation decision-makers must balance time and cost while considering external factors like the starting time.

Figure 12.

Comparison of solution results for adaptive differential evolution, genetic, and Dijkstra algorithms: (a) transportation time and (b) transportation cost.

Figure 13.

Average transportation time and cost for each month: (a) average transportation time and (b) average transportation cost.

Last but not least, to investigate the impact of uncertainty confidence level $α_{od}$ on experimental results, we varied the confidence level across 10 values ranging from 0.1 to 1. Figure 14, a and b , displays the comparison results for time and cost. The results show that, as the uncertainty confidence level increases, the total transportation time of the optimal route decreases, ensuring a well-met delivery deadline. However, the overall transportation cost exhibits an increasing trend. For instance, when $α_{od}$ is 0.1, the average transportation time over 30 days is 698 h, with a corresponding average transportation cost of $5,068. On the other hand, when $α_{od}$ is set to 1, the average transportation time over 30 days reduces to 561 h, but the corresponding average transportation cost increases to $5,374. In summary, higher confidence levels lead to shorter transportation times, meeting delivery deadlines more strictly, but at the expense of increased transportation cost. Conversely, at lower confidence levels, where delivery dates are well-met, the required transportation time is relatively longer, but the corresponding transportation costs are smaller. Additionally, variations in transportation time are linked to the distribution form and specific values of marine transportation delay time and multimodal transportation arrival dates. A larger interval in the distribution of total transportation time indicates stronger uncertainty, resulting in more noticeable changes in total transportation time with variations in the given confidence level. Conversely, smaller changes occur if the distribution interval is smaller. In practical applications, therefore, decision-makers should choose values based on specific circumstances.

Figure 14.

Comparison of route optimization results under different confidence levels: (a) total transportation time and (b) total transportation cost.

Conclusions

This paper focuses on the uncertainty of multimodal transportation time caused by marine transport delays, a common occurrence in international multimodal transport. The approach involves seeking reasonable transport routes based on the expected delay time in marine transportation to approximately estimate total transportation time. Ultimately, the goal is to establish an opportunity-constrained planning model for multimodal transportation route optimization, that takes into account the uncertainty of total transportation time. Utilizing knowledge from uncertainty theory, the model is transformed into a deterministic equivalent model, enhancing its effectiveness. Additionally, an ADE algorithm is developed to solve the model, and its performance is compared with GA and Dijkstra algorithm. The results indicate that:

The proposed model significantly reduces transportation time and cost, making it more realistic compared with deterministic models.

Differentiated optimal transportation routes can be obtained based on shipper-specific time and cost requirements. However, insufficient time or budget may result in the model having no solution. The optimal solution may remain unchanged as time and cost gradually relax.

Although transportation time and cost vary to some extent between different months, the overall optimal solutions obtained by the three algorithms perform best in May. This suggests that the start time of container transportation may influence the optimal transport route.

The ADE algorithm outperforms the GA and Dijkstra algorithm in solution efficiency.

The uncertainty confidence level is a crucial factor influencing the choice of transport routes. As the confidence level $α_{od}$ increases, the total transportation time of the optimal route decreases to ensure delivery deadlines are met, but total transportation costs increase.

Therefore, multimodal transportation route optimization involves a comprehensive problem with various factors, and decision-makers need to strike a balance between time and cost to obtain the safest transportation route.

Supplemental Material

sj-docx-1-trr-10.1177_03611981241242377 – Supplemental material for Research on Optimization of International Multimodal Transportation Routes Based on Uncertainty Theory

Supplemental material, sj-docx-1-trr-10.1177_03611981241242377 for Research on Optimization of International Multimodal Transportation Routes Based on Uncertainty Theory by Xiaoyang Wang, Peng Zhang, Zhigang Du, Ai-Qing Tian and Hongxia Lv in Transportation Research Record

Footnotes

Author Contributions

The authors contributions to the paper as follows: The authors' contributions to the paper are as follows: study conception and design: Xiaoyang Wang, Hongxia Lv; data collection: Peng Zhang, Zhigang Du, Ai-Qing Tian; analysis and interpretation of results: Xiaoyang Wang, Peng Zhang; draft manuscript preparation: Xiaoyang Wang. Supervision, validation, funding acquisition: Hongxia Lv. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Key R&D Program of China (2022YFB4300502); National Natural Science Foundation of China (Project Nos. 52072314, 52172321, 52102391); Key R&D Program of Guangzhou (202206030007); China State Railway Group Co., Ltd. Science and Technology Program (P2022X013; K2023X030); Key Science and Technology Projects in the Transportation Industry of the Ministry of Transport (2022-ZD7-131); the fundamental research funds for the central universities (2682022ZTPY068).

ORCID iD

Ai-Qing Tian

Data Accessibility Statement

Some or all data, models, or code generated or used during the study were provided by Southwest Jiaotong University under license, therefore, cannot be made freely available. Direct requests for these data should be made to Southwest Jiaotong University.

Supplemental Material

Supplemental material for this article is available online.

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