
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12326-2
10.1016/j.heliyon.2024.e36295
e36295
Research Article
Connectivity reliability evaluation and most reliable shipping route choice in a seaborne crude oil network
Wang Shuang wangshuang91@dlmu.edu.cn
a⁎
Wang Yan b
Lai Chengshou c
a College of Transportation Engineering, Dalian Maritime University, Dalian, 116026, China
b CSSC Systems Engineering Research Institute, Beijing, 100094, China
c CCCC Water Transportation Consultants Co., Ltd., Beijing, 100007, China
⁎ Corresponding author. Linghai Road No. 1st, Ganjingzi District, Dalian, 116026, Liaoning Province, China. wangshuang91@dlmu.edu.cn
14 8 2024
30 8 2024
14 8 2024
10 16 e362959 5 2024
23 7 2024
13 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
The connectivity reliability of strait and canal nodes in seaborne crude oil networks is uncertain because of various risk factors. Existing studies have mainly focused on road networks and often ignored real-world factors by assuming fixed and identical values for the connectivity reliability of each node, leading to inaccurate estimations. Few studies have considered node reliability when identifying the most reliable routes for oil shipments in response to various external and changing risks. To address these limitations, we first establish new connectivity reliability evaluation methods for both nodes and networks. Then, we develop the α-most reliable shipping route and the very most reliable shipping route models using uncertain programming to dynamically identify the most reliable routes for crude oil, ensuring timely and safe transportation. We apply these models to China's seaborne network of imported crude oil. The results show a network connectivity reliability of 0.6228, which is impacted by unreliable origin–destination pairs in the Middle East. The risk values of the most reliable oil shipping routes vary regionally, with higher values in Africa and the Middle East than in Asia and Latin America. As node risk increases, regional disparities also increase. These findings will aid in the development of energy transportation and import strategies to enhance transportation reliability.

Keywords

Connectivity reliability
Seaborne network
Crude oil shipments
Most reliable oil shipping route
==== Body
pmc1 Introduction

Oil is a vital resource for the military, economic, and social development of all countries and is often likened to the “lifeblood of industry.” However, the distribution of oil resources is geographically and spatially imbalanced, leading to disparities between international oil supply and demand [1]. Shipping is the predominant mode of transportation for crude oil, accounting for approximately 90 % of the world's total crude oil trade [2]. The safety and reliability of oil shipping have significant impacts on the energy and economic security of nations.

The critical strait and canal nodes used for seaborne crude oil transportation are highly susceptible to risk factors such as ship accidents, piracy, unpredictable weather patterns, and geopolitical tensions [[3], [4], [5], [6]]. The current tense situation in the Red Sea region presents significant reliability risks for oil shipments passing through the Suez Canal and Bab-el-Mandeb Strait. The ongoing conflict between Russia and Ukraine also threatens the safety of vessels passing through the Strait of Bosporus. These risk factors contribute to the indeterminacy of the connectivity reliability at each node. Disruptions to nodes can compromise the timely delivery of crude oil. Therefore, evaluating the connectivity reliability of seaborne networks and selecting the most reliable routes for oil shipments, while considering various risk factors, is crucial.

Previous studies have predominantly focused on the connectivity reliability of land transportation networks such as urban road networks [7,8]. However, the connectivity reliability of seaborne networks has not yet been thoroughly investigated. Limited studies, such as those by Wang et al. [2] and Xiao et al. [9], have assumed that the connectivity reliability of maritime nodes (straits, canals, and ports) is a known quantity and have proceeded to assess the overall network reliability. Nevertheless, the evaluation of connectivity reliability in maritime nodes is yet to be sufficiently addressed, particularly given the potential for uncertain disruptions. Several other studies have focused solely on port reliability [10] or examined the impact of port disruptions on the configuration of the maritime network [11], without considering the reliability of the strait and canal nodes or the entire maritime network. Additionally, in the selection of crude oil shipping routes, shipping expenses are a crucial consideration; however, connectivity reliability is often overlooked [12,13]. In this study, we aim to address these gaps in the literature.

We propose connectivity reliability evaluation models and the most reliable oil shipping route selection model for seaborne oil networks. Several challenges must be addressed to achieve this goal. (1) Research on connectivity reliability evaluation of seaborne networks is in its infancy. Existing studies assumed a fixed and identical value for the connectivity reliability of each node [2], and failed to consider real-world factors. (2) Maritime nodes are subject to a multitude of influences, including geopolitical stability, environmental conditions, and non-traditional risks such as piracy and maritime terrorism [14]. The complexity of these risk factors makes it challenging to evaluate the connectivity reliability of maritime nodes. (3) The available data on disruptions to maritime nodes resulting from various risk factors are insufficient, making it difficult to ascertain the most reliable oil shipping routes, especially when accounting for dynamic and unpredictable risks.

The main contributions of this study are as follows. First, to address the limitations of assuming fixed and identical values for each node's connectivity reliability, we propose evaluation models for maritime nodes and an entire seaborne crude oil network based on the uncertainty theory. This provides a more accurate characterization of both node and network reliability. Second, we develop the α-most and very most reliable shipping route models to dynamically identify the most reliable routes, surpassing the traditional cost-based approach for oil shipments facing external changing risks. Third, we apply these models to Chinese import data. The results, based on the varying risk values for each node, can help formulate more effective strategies for oil shipments and imports.

The remainder of this paper is organized as follows. Section 2 reviews relevant literature. Section 3 introduces the concepts of uncertainty theory used in this study. Section 4 describes the problem and presents our proposed methodology. A case study is presented in Section 5, and the conclusions and implications are discussed in Section 6.

2 Literature review

ScienceDirect, Web of Science, and IEEE Xplore are the primary databases used in the literature searches. The first stream pertains to the evaluation of connectivity reliability in transportation networks. Japanese scholars introduced this concept in 1982, focusing on the probability that at least one path between two nodes remains connected. They focused on the binary states of road segments, considering them to be either connected or not connected [7]. Subsequently, this concept was extended to evaluate the connectivity reliability between multiple nodes (k nodes) and the entire network [15].

The connectivity reliability of transportation networks is primarily analyzed using analytical and approximation methods [16]. Analytical methods typically involve the use of minimal paths and cut set algorithms [[17], [18], [19], [20]]. To reduce the calculation time, the disjoint minimal path set algorithm, which is defined as the sum of irrelevant minimal paths, is employed [21]. However, the computational complexities of these classic algorithms increase exponentially with the number of nodes in a network. To improve the computational efficiency, researchers have proposed network decomposition algorithms [16,[22], [23], [24]]. For approximation, Monte Carlo simulation is a common approach [25,26]. Innovative approaches have also been introduced [[27], [28], [29]]. For instance, Dai et al. [30] used an effective path number measurement based on complex network theory to assess the connectivity reliability of highway networks using Shandong Province as a case study under various scenarios. Jiang and Huang [31] proposed a Bayesian network model to evaluate connectivity reliability in high-seismicity regions and identify critical units within highway networks. Although these studies yielded valuable insights into the reliability of network connectivity, they were largely confined to examining land networks. Although studies on maritime risk assessment techniques, such as machine learning, have garnered increasing attention [[32], [33], [34], [35]], research on the connectivity reliability of seaborne networks remains limited.

The second stream of literature addresses the reliable route choice problem in transportation networks. In recent decades, numerous studies have focused on ship scheduling and routing for crude oil transportation, with an emphasis on factors such as shipping expenses and environmental impacts [12,[36], [37], [38]]. However, there is a growing recognition that transportation links or nodes within the network may become disconnected owing to risk events, highlighting the importance of identifying the most reliable route. Reliable route choices in transportation networks have garnered increasing attention in recent years [[39], [40], [41], [42]].

Some scholars defined the reliable routing problem as the selection of the route with the highest level of reliability from a given origin and destination, considering potential disruptions [39,41]. Rashidian et al. [39] employed failure mode and effect analysis methods and consulted experts to evaluate the overall reliability of the network. Subsequently, the reliability of the nodes and arcs is assigned, and a reliable route is selected for goods transportation. Hosseini [41] proposed a path choice model with the objective of identifying routes with maximum reliability for post-disaster transportation networks based on uncertainty theory, facilitating the safe delivery of supplies to affected regions. In addition, several studies have focused on transportation time reliability to address the issue of reliable route choices in the context of network uncertainty [[43], [44], [45]]. Under these circumstances, the reliable route choice problem is defined as the identification of routes with a minimum transportation time that satisfy a given reliability constraint [45,46], the maximum probability of achieving on-time arrival within a given time budget [47], or the minimum time budget that can be achieved with a given probability [44,48]. For example, Chen et al. [45] developed an efficient path-finding algorithm to determine the shortest reliable path in stochastic and time-dependent networks with spatiotemporally correlated link travel times and conducted a case study on the road network in Beijing. Xing et al. [46] developed a collaborative reliable route choice model for truck-drone applications with the objective of minimizing total travel time and associated variabilities. However, many of these studies have failed to consider the reliability of connectivity, which is of paramount importance for guaranteeing uninterrupted network performance, even under challenging circumstances. Moreover, research on identifying the most reliable routes in seaborne networks is limited.

In this study, we introduce a novel connectivity reliability evaluation method using the uncertainty theory and a disjoint minimal path set algorithm, specifically tailored for maritime nodes and oil shipping networks. This study represents one of the first attempts to systematically assess the connectivity reliability of individual nodes in maritime transportation. Additionally, we propose a reliable route choice model for oil shipping networks that considers the connectivity reliability of each node, which has been largely overlooked in previous studies. We apply our model to the imported crude oil seaborne network of China, which is the world's largest oil importer. The results of our analysis offer practical and significant policy recommendations for enhancing transportation reliability and optimizing oil shipment strategies.

3 Preliminaries

In this section, we introduce the fundamental concepts of uncertainty theory, including the notions of uncertain measure, uncertain variable, uncertainty distribution, and the α-pessimistic value of uncertain variables.

Let Θ be a nonempty set and L be a σ− algebra over Θ. Each element Λ∈L is designated as a number M{Λ}, a set function M that satisfies the following axioms is defined as an uncertain measure, and the triplet (Θ,L,M) is referred to as an uncertainty space [49].(1) M{Θ}=1.

(2)For any event Λ, M{Λ}+M{Λc}=1.

(3)For every countable sequence of events {Λi}, M{∪i=1∞Λi}≤∑i=1∞M{Λi}.

(4)If Λ1⊆Λ2, M{Λ1}≤M{Λ2}.

Definition 1 An uncertain variable η is defined as a measurable function from the uncertainty space Θ,L,M to a set of real numbers, R [50].

Definition 2 For any real number x∈R, the uncertainty distribution characterizing the uncertain variable η is expressed as Γx=Mη≤x [40].

The inverse function Γ−1 is defined as the inverse uncertainty distribution function of η. If Γ−1 exists and is unique, the uncertainty distribution is designated as regular.

Definition 3 Let α∈(0,1], and for any real number x∈R, ηinf(α)=inf{x|M{η≤x}≥α} is defined as the α-pessimistic value of the uncertain variable η, that is, the α-pessimistic value is equal to the minimum value of x that satisfies the condition M{η≤x}≥α.

The α-pessimistic value can be derived from the inverse uncertainty distribution of the uncertain variable η, which can be expressed as ηinf(α)=Γ−1(α).

Theorem 1 Let η1,η2,…,ηn be independent uncertain variables with uncertainty distribution functions Γ1,Γ2,…,Γn, respectively. If f is a strictly increasing function, then η=f(η1,η2,…,ηn) is an uncertain variable [51].

Theorem 2 Suppose η1,η2,ηi…,ηn denote independent uncertainty variables with regular uncertainty distributions specified by Γ1,Γ2,…,Γn. If f(x1,x2,…,xn) is a continuous and strictly increasing function over x1,x2,…xk and is a continuous and strictly decreasing function over xk+1,xk+2,…,xn, then the uncertainty variable f(ηP)=f(η1,η2,…ηn) has an inverse uncertainty distribution [50]:(1) Ψ−1(α)=f(Γ1−1(α),…,Γi−1(α),Γi+1−1(1−α),…,Γn−1(1−α))

where Γi−1 is the inverse uncertainty distribution of ηi.

4 Methodology

In this section, we propose a research framework for evaluating connectivity reliability and the choice of the most reliable shipping route in a seaborne crude oil network, as illustrated in Fig. 1.Fig. 1 Research framework.

Fig. 1

In this framework, uncertain variables are first employed to describe the connectivity reliability of maritime nodes, and the disjoint minimal path set algorithm is then applied to evaluate the connectivity reliability of each origin–destination (OD) pair. Subsequently, the overall reliability of the network is evaluated using the weighted average of the connectivity reliability across all OD pairs. Second, the most reliable oil shipping route is dynamically identified based on the connectivity reliability of the node. This is achieved by solving the α-most and very most reliable oil shipping route problems, which are formulated using uncertain programming.

4.1 Problem description

In this study, we define S=(T,A) to represent the global seaborne crude oil network, where T and A denote the sets of nodes and legs, respectively, as depicted in Fig. 2. The connectivity reliability of straits, canals, and ports under various risk factors, such as piracy, shipping accidents, unforeseen weather patterns, and geopolitical tensions, cannot be definitively determined. Given the lack of disruption data for nodes to estimate their probability distribution, traditional random variables may not effectively describe these uncertainties. Uncertainty theory, however, has been validated as an effective alternative. Therefore, we employ uncertainty theory to address the uncertain factors within the seaborne network [51,52].Fig. 2 Global seaborne crude oil network.

Fig. 2

The uncertainty variable ηi ((0<ηi<1)) is associated with each maritime node and characterizes its connectivity reliability. The variable is commonly used across different domains to represent uncertain quantities such as market supply, demand, transportation cost, and capacity [53,54]. In this paper, we aim to investigate the connectivity reliability of different nodes and identify the most reliable shipping route within a crude oil seaborne network affected by various risk factors.

Each maritime node is categorized into one of two states: operational at full capacity and capable of handling its maximum capacity; or in a state of disruption and unable to contribute any capacity to the network. This characterization assumes that the legs between nodes function normally and that each node operates autonomously without being dependent on other nodes for its functionality.

4.2 Connectivity reliability evaluation

When a maritime node is affected by risk events, the reliability of the shipping route in which the node is located may be disrupted. The connectivity reliability of an OD pair is defined as the probability that the OD pair remains connected, despite some nodes being exposed to risk events within the seaborne network. The overall connectivity reliability of the seaborne crude oil network is calculated as the weighted average of the connectivity reliability of all OD pairs, with the weights determined by the proportion of the oil volume transported by each OD pair.

In this paper, we propose a novel approach to calculate the connectivity reliability of a seaborne oil network by combining minimal path sets with disjoint algebra. The traditional minimum path algorithm is commonly used to calculate network reliability. However, it suffers from high computational complexity owing to the large number of minimum paths it generates. To address this challenge, we introduce a disjoint minimal path set algorithm. Let P1, P2, …, Pk denote all minimal paths of a given OD pair in the seaborne oil network. The connectivity reliability of the OD pairs can be expressed as follows:(2) Ro,d=Pr(P1+P2+⋅⋅⋅+Pk)

Based on disjoint theory, P1+P2+⋅⋅⋅+Pk can be computed as(3) P1+P2+⋅⋅⋅+Pk=P1+P1‾P2+P1‾P2‾P3+⋅⋅⋅+P1‾P2‾⋅⋅⋅Pk−1‾Pk

The items in equation (3) are disjointed. Therefore, Ro,d can be expressed as(4) Ro,d=Pr(P1)+Pr(P1‾P2)+Pr(P1‾P2‾P3)+⋅⋅⋅+Pr(P1‾P2‾⋅⋅⋅Pk−1‾Pk)

According to the sum of the disjoint product theorem, if Pj and Ph are two paths that share common nodes, Pj‾Ph=Pj−Ph‾Ph. Pj−Ph‾ denotes the Boolean product of the nodes contained in Pj but not in Ph.

Based on the above theorem, equation (4) can be simplified as follows:(5) Ro,d=Pr(P1)+Pr(P1‾P2)+Pr(P1‾P2‾P3)+⋅⋅⋅+Pr(P1‾P2‾⋅⋅⋅Pk−1‾Pk)=Pr(P1)+Pr(P1−P2‾P2)+Pr(P1−P3‾P2−P3‾P3)+⋅⋅⋅+Pr(P1−Pk‾P2−Pk‾⋅⋅⋅Pk−1−Pk‾Pk)=Pr(P1)+Pr(P1−P2‾)Pr(P2)+Pr(P1−P3‾P2−P3‾)Pr(P3)+⋅⋅⋅Pr(P1−Pk‾P2−Pk‾⋅⋅⋅Pk−1−Pk‾)Pr(Pk)

The connectivity reliability of any shipping route in equation (4) is a function of the reliability of its constituent nodes. In particular, the reliability of route Pj can be expressed as(6) Pr(Pj)=∏i=1nηi

where n denotes the number of nodes on route Pj. The connectivity reliability of each maritime node is characterized by a regular uncertainty distribution represented by Γ.(7) Γ(x)=M{ηi≤x},∀x∈R

where M is an uncertainty measure, and R is the set of real numbers.

Unlike ordinary real variables, the connectivity reliability of maritime nodes cannot be directly compared and calculated. To facilitate this, we propose α-pessimistic values as critical thresholds for decision-making, aiding in comparison and computation. Here, given a confidence level α∈(0,1], the associated α-pessimistic value of ηi is computed as(8) ηinf(α)=inf{x|M{ηi≤x}≥α}

This value can be obtained from the inverse uncertainty distribution of the node's connectivity reliability, denoted by Γ−1.(9) ηinf(α)=Γ−1(α)

The connectivity reliability of each OD pair can be evaluated according to equations (5), (6), using the α-pessimistic value of each maritime node's connectivity reliability, which is calculated using equation (9). Subsequently, the connectivity reliability of the entire seaborne network can be calculated based on the proportion of crude oil imports transported by each OD pair.

4.3 Most reliable oil shipping route choice

Considering the potential risk factors, we aim to identify the shipping route with the highest reliability for transporting crude oil in a timely manner, which is essential for ensuring the smooth operation of national economies. Owing to the indeterminate reliability of connectivity for maritime nodes, traditional methods are inadequate for addressing this problem. Therefore, we employ uncertainty programming to identify the most reliable shipping route in the seaborne oil network.

Let o∈T,d∈T represent the source and sink nodes, respectively. Our goal is to identify the most reliable shipping route for crude oil shipments, which is equivalent to maximizing the objective function ∏i∈Pηi over a set of all routes from o to d in the seaborne network. For simplicity, we convert the problem of finding routes with the highest reliability into a risk-minimization task by inverting the reliability variables. The inverted variables are denoted by ηi′, which represent the uncertainty risk associated with the maritime nodes. Let ρ={Pk} denote the set of all o-d routes in the seaborne network. Shipping route P in the seaborne network, characterized by the positive risk variables ηi′, can be depicted as ηp={ηi′|i∈P}. The risk value for route P can be calculated as follows:(10) f(ηp)=∏i∈P(1−xi+ηi′xi)

where xi = 1 if node i is part of route P and zero otherwise.

According to Theorem 1, the value of f(ηp) is uncertain. To determine the oil shipping routes with the highest reliability, f(ηp) should be minimized.(11) minf(ηp)=min∏i∈P(1−xi+ηi′xi)

To address the challenge of directly minimizing equation (11), we employ uncertainty programming that leverage critical values of risk variables and introduce the concept of the α-most reliable oil shipping route. The α-pessimistic value, chosen as the critical value, is instrumental in solving the problem of choosing the most reliable oil shipping route. Based on the definition of the α-pessimistic value and equation (8), the α-pessimistic value of ηp can be expressed as(12) ηPα=inf{r|M{f(ηP)≤r}≥α}

where α∈(0,1] is the predefined confidence level, and r denotes the risk value.Definition 4 The α-most reliable oil shipping route: Defined as the oil shipping route with the lowest risk value, given a specified confidence level α. For any other oil shipping route P* from o to d, if ηp*α≤ηpα, route ηp* is referred to as the α-most reliable oil shipping route. Specifically,(13) inf{r|M{f(ηP*)≤r}≥α}≤inf{r|M{f(ηP)≤r}≥α}

Therefore, the model for identifying the most reliable oil shipping route is converted to minimize the value of r:(14) minr∈R,P∈ρr

(15) M{∏i∈Pηi′≤r}≥α

As ηi′ has a regular uncertainty distribution and f(ηp) is continuous and strictly increasing, f(ηP*)=∏i∈P*ηi′ has an inverse uncertainty distribution, Ψ′−1(α). In addition, f(ηP*) satisfies(16) M{f(ηP*)≤Ψ′−1(α)}=α

Based on the definition of the α-pessimistic value, for any maritime node i ∈ T and α∈(0,1), the α-pessimistic value of ηi′, which is denoted as η′iα, can be calculated as η′iα=Γ′i−1(α), where Γ′i−1 represents the inverse uncertainty distribution of ηi′.

Aligning with the monotonicity theorem in the uncertainty theory, for each shipping route in the seaborne crude oil network, the following expression can be formulated:(17) M{∏i∈Pηi′≤∏i∈Pη′iα}≥M{∩{ηi′≤η′iα}}=minM{ηi′≤η′iα}=min{α}=α

In addition, given that f(ηP*) is strictly increasing and that the ηi′ values are independent of each other,(18) M{∏i∈Pηi′≤∏i∈Pη′iα}≤M{∪{ηi′≤η′iα}}=1−M{∩{ηi′>η′iα}}=1−minM{ηi′>η′iα}=1−min{1−α}=α=minM{ηi′≤η′iα}=min{α}=α

Equations (17), (18) yield(19) M{∏i∈Pηi′≤∏i∈Pη′iα}=α

According to equation (16), ∏i∈Pη′iα represents the inverse uncertainty distribution and the α-pessimistic value of f(ηP*). Therefore, the constraint in expression (15) can be substituted with M{∏i∈Pηi′≤r}=α. Constraint M{∏i∈Pηi′≤∏i∈Pη′iα}≥α is valid if and only if ∏i∈Pη′iα≤r is valid. Minimizing r in expression (14) is then equivalent to the minimization of ∏i∈Pη′iα.(20) minP∈ρ∏i∈Pη′iα

In this paper, the risk variable for each maritime node is defined as the inverse of its connectivity reliability and represented as ηi′=1/ηi. According to Theorem 2 and Equation (1), for any node in the seaborne oil network, given α∈(0,1],(21) η′iα=Γ′i−1(α)=f(Γi−1(1−α))=1Γi−1(1−α)

In summary, within the seaborne crude oil network affected by various risk factors, each maritime node's risk value is determined by α. By adjusting α, the most reliable oil shipping route can be dynamically identified based on equations (20), (21). The decision-making process is illustrated in Fig. 3.Fig. 3 Decision-making process.

Fig. 3

Definition 5. The very most reliable oil shipping route: Occasionally, the decision maker may specify a risk value criterion and identify a shipping route for transporting crude oil that remains below this risk threshold. Therefore, the concept of the very most reliable oil shipping route is introduced in this paper to identify the route that offers the highest probability of satisfying the specified risk requirement.

For any oil shipping route P∈ρ from o to d, if(22) M{f(ηP*)≤r}≥M{f(ηP)≤r}

P* is the very most reliable shipping route in the seaborne oil network.

Let Ψ′ represent the uncertainty distribution of the most reliable shipping route in the seaborne oil network and let r be the predetermined risk value. Then, the problem of identifying the very most reliable oil shipping route can be addressed by solving the α-most reliable oil shipping route problem by assigning α=Ψ′(r) [50], as shown in Fig. 3.

5 Case study

In this section, we present a case study of a Chinese imported crude oil seaborne network. We apply the proposed model to evaluate the connectivity reliability of China's seaborne crude oil network considering various risk factors. Subsequently, the α-most and very most reliable oil shipping routes are identified based on the evaluation results.

5.1 Problem setting

According to the Trade Statistics Database of the International Trade Center (ITC, https://intracen.org/resources/data-and-analysis/trade-statistics), in 2022, China imported 508 million tons of crude oil, with the Middle East, Africa, and Latin America serving as the top three source regions, accounting for 52.7 %, 10 %, and 7.3 % of the total imports, respectively. In addition, approximately 7.2 % and 1.6 % of crude oil originated from Southeast Asia and Europe, respectively. Fifteen main source countries are selected for analysis. Fig. 4 depicts the straits and canals used for oil shipments imported from these countries along with the corresponding seaborne crude oil network. Collectively, these countries contributed to 74.3 % of China's total crude oil imports. One representative loading port is chosen for each source country because of its similar exposure to risk factors. The Ningbo Port is selected as the unloading port for China [55].Fig. 4 China's seaborne crude oil network.

Fig. 4

The reliability of connectivity for each maritime node, with the exception of Ningbo Port, follows a zigzag uncertainty distribution represented by η∼Ζ(a,b,c), whose inverse distribution is expressed by equation (23), where a<b<c. The values of a, b, and c for the straits and canals are derived based on security assessment scores [55], as illustrated in Fig. 5. For port nodes, the values of a, b, and c are derived from the World Bank's Ease of Doing Business Ranking, as depicted in Table 1. Given a specified confidence level α, according to equations (21), (23), the connectivity reliability and risk of each maritime node can be calculated.(23) Γ−1α={1−2αa+2αb21−αb+2α−1cifα<0.5ifα≥0.5

Fig. 5 The values of a, b, and c for straits and canals (represented by the numbers in parentheses).

Fig. 5

Table 1 Values of a, b, and c for port nodes in source countries and the import share of source countries.

Table 1Region	Port node	a,b,c	Import share	
Middle East	Saudi Arabia	0.7,0.75,0.85	17.21 %	
Iraq	0.65, 0.73, 0.78	10.92 %	
UAE	0.8, 0.88, 0.95	8.41 %	
Kuwait	0.7, 0.73, 0.82	6.55 %	
Oman	0.72, 0.75, 0.78	7.75 %	
Africa	Angola	0.65, 0.71, 0.82	5.92 %	
Congo	0.65, 0.68, 0.82	1.40 %	
South Sudan	0.6, 0.7, 0.75	0.06 %	
Libya	0.6, 0.68, 0.75	0.74 %	
Latin America	Colombia	0.8, 0.82, 0.9	1.70 %	
Brazil	0.68, 0.75, 0.8	4.90 %	
Asian	Malaysia	0.84, 0.88, 0.92	7.02 %	
Indonesia	0.7, 0.75, 0.82	0.15 %	
Europe	Norway	0.88, 0.9, 0.95	1.18 %	
UK	0.9, 0.92, 0.95	0.44 %	
Note: data source for the import share: International Trade Center.

5.2 Results

5.2.1 Connectivity reliability

Taking the confidence level as 0.9, the connectivity reliability of each OD pair is evaluated based on the model proposed in Section 4.2. Considering the share of oil transported by each OD pair, the overall connectivity reliability of the entire seaborne crude oil network is calculated to be 0.6228. This indicates that the seaborne network can handle approximately 60 % of the imported crude oil transport tasks at the current level of node connectivity reliability. The observed level of connectivity reliability is relatively low, primarily because of the lower connectivity reliability of OD pairs in the Middle East, which accounts for a significant share of crude oil shipments. The connectivity reliability of the regional seaborne oil network is similarly assessed, as depicted in Fig. 6.Fig. 6 Connectivity reliability of the regional seaborne oil network.

Fig. 6

The regional seaborne network for crude oil imports from the Middle East and Africa to China exhibits comparatively low connectivity reliability, whereas networks from Latin America, Asia, and Europe to China are more reliable. This difference is primarily due to the challenges faced by most OD pairs in the Middle East, which must pass through risky maritime nodes such as the Hormuz and Malacca Straits. In addition, some OD pairs in Sudan are transported through the Bab-el-Mandeb Strait, and some in Libya use the Bab-el-Mandeb Strait and Suez Canal, further affecting reliability. The connectivity reliability of the regional network from Africa to China is hindered by higher political risks in the exporting countries and the higher risks associated with the Bab-el-Mandeb and Suez Canal routes. By contrast, certain Latin American oil imports can be shipped to China via the Panama Canal, avoiding the Strait of Malacca. Imported crude oil from Southeast Asia can also be transported directly to China via the Taiwan Strait. Furthermore, in Europe, the political risk in the exporting countries is relatively low, and OD pairs are allowed to avoid the Strait of Hormuz, thus contributing to higher connectivity reliability.

5.2.2 Most reliable oil shipping route

(1) α-most reliable oil shipping route

Because similar nodes exist for each OD pair within the same region, the OD pairs within the same region can be compared based on their connectivity reliability. When considering each region as a whole, identifying the α-most reliable oil shipping route for each region holds more practical significance than focusing on individual OD pairs within the region. In this paper, five virtual nodes, namely, S1, S2, S3, S4, and S5, are introduced as starting nodes representing the Middle East, Africa, Latin America, Southeast Asia, and Europe, respectively. Assuming that the connectivity reliability of these five virtual nodes is 1 and that all other factors are equal, the α-most reliable oil shipping routes for the five regions can be determined. Table 2 presents the α-most reliable oil shipping routes and their corresponding risks for the five regions when α = 0.9.Table 2 α-most reliable oil shipping route of each region.

Table 2Source region	Most reliable oil shipping route	Risk	
Middle East	S1-Oman-Lombok Strait-Taiwan Strait-Ningbo	2.47	
Africa	S2-Angola-Lombok Strait-Taiwan Strait-Ningbo	2.71	
Latin America	S3-Colombia-Panama Canal-Ningbo	1.63	
Southeast Asia	S4-Malaysia-Taiwan Strait-Ningbo	1.50	
Europe	S5-UK-Lombok Strait-Taiwan Strait-Ningbo	1.99	

For each region, a path is selected at a given confidence level of 0.9, indicating that the path's risk value is within a certain range with a probability of at least 0.9. A lower risk value is preferred for these selections. The chosen routes for different regions typically involve navigating through a limited number of chokepoints and bypassing relatively risky straits such as the Hormuz Strait, Bab-el-Mandeb, Suez Canal, and Malacca Strait. Specifically, in the Middle East, sailing from Oman through the Lombok Strait and then through the Taiwan Strait to the Port of Ningbo is preferable to avoid riskier routes. In Africa and Europe, to avoid going through the Suez Canal and Bab-el-Mandeb, the route from Angola or the UK through the Lombok and Taiwan Straits is considered the most reliable. Similarly, for Latin America, traveling through fewer nodes, such as crossing the Pacific to China via the Panama Canal, thereby avoiding the Strait of Malacca, is preferable for reducing the risks associated with oil shipments.

In addition, the number of nodes involved in oil shipping routes through the Middle East and Africa tends to be greater than those through Latin America and Southeast Asia. Despite efforts to circumvent the Hormuz Strait and Suez Canal, the nodes encountered along the Middle East and Africa routes often pose higher risks. Consequently, the risks associated with the α-most reliable oil shipping routes for the Middle East and Africa are typically higher than those associated with Latin America and Southeast Asia. Importing more crude oil from Latin America and Southeast Asia may be beneficial for addressing this disparity and enhancing the reliability of oil shipments.

Recall that, based on equation (21), the risk value of each maritime node is determined by the confidence level α. The flexibility of our model enables decision-makers to assess route options across various values of α, allowing for the dynamic identification of the most reliable oil shipping route in response to evolving international situations and fluctuating risks at maritime nodes. Table 3 illustrates how different values of α (for example, 0.8, 0.5 and 0.1) impact the choice of the most reliable oil shipping routes.Table 3 The most reliable oil shipping routes for different α values.

Table 3α	Most reliable oil shipping route	r	
0.80	S1-Oman-Lombok Strait-Taiwan Strait-Ningbo	2.41	
S2-Angola-Lombok Strait-Taiwan Strait-Ningbo	2.62	
S3-Colombia-Panama Canal-Ningbo	1.59	
S4-Malaysia-Taiwan Strait-Ningbo	1.48	
S5-UK-Lombok Strait-Taiwan Strait-Ningbo	1.94	
0.50	S1-Oman-Sunda Strait-Taiwan Strait-Ningbo	2.08	
S2-Angola-Sunda Strait-Taiwan Strait-Ningbo	2.20	
S3-Colombia-Panama Canal-Ningbo	1.49	
S4-Malaysia-Taiwan Strait-Ningbo	1.42	
S5-UK-Sunda Strait-Taiwan Strait-Ningbo	1.70	
0.20	S1-UAE-Hormuz Strait-Sunda Strait-Taiwan Strait-Ningbo	1.90	
S2-Angola-Sunda Strait-Taiwan Strait-Ningbo	1.90	
S3-Colombia-Panama Canal-Ningbo	1.34	
S4-Malaysia-Taiwan Strait-Ningbo	1.33	
S5-UK-Sunda Strait-Taiwan Strait-Ningbo	1.57	

The choice of the most reliable oil shipping route across regions such as the Middle East, Africa, and Europe is notably influenced by variations in α. For the Middle East, when the risk in the Strait of Hormuz is high or medium (α = 0.8 or 0.5), the most reliable oil transportation route is from Oman to Ningbo via the Lombok or Sunda Straits and the Taiwan Strait. This choice is driven by the desire to avoid higher risks associated with the Strait of Hormuz during periods of elevated or moderate risk. Therefore, choosing a route that bypasses this strait (through the Lombok or Sunda Straits) is considered more reliable under these circumstances. Conversely, when the risk in the Strait of Hormuz is low (α = 0.2), the most reliable oil route shifts to transporting oil from the UAE to Ningbo via the Strait of Hormuz and Sunda Strait. In this scenario, the reduced risk in the Strait of Hormuz makes the direct route from the UAE through this strait more appealing and reliable than routes that bypass it.

In the context of Africa and Europe, when both the Sunda and Lombok Straits are deemed dangerous (α = 0.8), the most reliable option for crude oil shipments is through the Lombok Strait. This decision is driven by the relative risk perception that the Sunda Strait presents a higher value of risk than the Lombok Strait under these conditions. Therefore, the Lombok Strait route is considered safer and more reliable. Conversely, when the risk values of the Lombok and Sunda Straits are low or medium (α = 0.2 or 0.5), the most reliable option shifts to transporting crude oil through the Sunda Strait. In this scenario, the Sunda Strait is perceived to have a lower risk value than the Lombok Strait, making it a preferred and more reliable route under conditions of reduced risk.(2) Very most reliable oil shipping route

The objective of identifying the very most reliable oil shipping routes based on a specified risk threshold is to ensure that the risk of the oil shipping route remains below the threshold with the highest probability. Given the risk values provided, that is, rs1 = 2.08, rs2 = 2.20, rs3 = 1.49, rs4 = 1.42, and rs5 = 1.70, as shown in Table 3, ΨS1′(2.08)=0.5, ΨS2′(2.20)=0.5, ΨS3′(1.49)=0.5, ΨS4′(1.42)=0.5, and ΨS5′(1.70)=0.5 can be obtained, and the very most reliable routes can then be identified. For example, in the Middle East, the identified oil shipping route that satisfies the maximum likelihood requirement with the highest reliability is the S1–Oman–Sunda Strait–Taiwan Strait–Ningbo route.

Therefore, for any given risk value, by setting α=Ψ′(r) and identifying the corresponding α-most reliable oil shipping routes at different confidence levels, we can determine the very most reliable oil shipping routes.

Furthermore, by setting different confidence levels, Ψ′−1(α) can be calculated and the numerical uncertainty distribution of the most reliable oil shipping route f(ξP) can be determined, as illustrated in Fig. 7.Fig. 7 Uncertainty distribution of f(ξP).

Fig. 7

As shown in Fig. 7, for the same confidence level, the risk values associated with the most reliable oil shipping routes are relatively higher in Africa and the Middle East than in Asia and Latin America. However, as the confidence level α increases, indicating a higher perception for risk, the disparities between the risks of the most reliable oil shipping routes in these regions become more pronounced. In the event of deterioration in the international situation and increased risks associated with certain maritime nodes, it would be prudent to import more crude oil from Asia and Latin America. This strategic shift aims to mitigate risk by relying on routes with lower associated risk values during periods of heightened instability. Conversely, during periods of relative international stability and reduced risk to nodes, increased imports from the Middle East could be advantageous owing to the shorter transport distances involved.

5.3 Discussion

Currently, the issue of connectivity reliability in seaborne networks has been addressed in only a limited number of studies. These studies employed a simulation approach using random and intentional attacks on nodes to set fixed values for the connectivity reliability of each maritime node [2,9]. Specifically, if a node is attacked, its connectivity reliability is set to zero; otherwise, it is set to one or a fixed value. The network reliability is then assessed under these attack scenarios [2]. However, it is unrealistic to assume that multiple nodes will be attacked simultaneously, and assuming identical reliability for all unattacked nodes introduces bias into the evaluation outcomes. This paper employs uncertainty variables to characterize the distinct connectivity reliability of each node under various risk factors and examines their collective impact on network reliability. The results indicate that the network's overall connectivity reliability is relatively low, primarily due to unreliable OD pairs in the Middle East, as evidenced by the prevailing political instability in the region. An effective evaluation of connectivity reliability allows decision makers to improve their strategies and policies, thereby enhancing transportation reliability.

In addition, we investigate the dynamic identification of the most reliable oil shipping routes. Most existing studies prioritized cost considerations [12,13]. However, owing to the pivotal role of oil in the global economy, enhancing transportation reliability is crucial. The international situation is marked by high complexity and volatility, as evidenced by the Russia–Ukraine conflict and the ongoing Red Sea crisis. Our model enables identification and adjustment of shipping routes in response to geopolitical developments and variations in node reliability.

6 Conclusions and implications

In this paper, we propose connectivity reliability evaluation methods and the most reliable route choice model for crude oil seaborne networks. These methods are applied to China's seaborne network of imported crude oil. Our findings highlight that the network's connectivity reliability is relatively low, primarily because of the significant import share and relatively low connectivity reliability of OD pairs in the Middle East. The risk values of the most reliable oil shipping routes exhibit regional variations, with Latin America and Southeast Asia demonstrating relatively low risk values compared to those observed in the Middle East and Africa. As the international situation evolves and node risk values fluctuate, our model enables the identification and adjustment of the most reliable oil shipping routes by modifying the confidence level α.

The results of this study have several important implications. First, adaptive transportation strategies that respond to the dynamic international conditions and node-specific risks are required. For instance, in regions such as Africa and Europe, where certain straits, such as Sunda and Lombok, may pose higher risks, it is crucial to adjust shipping routes accordingly. When the risk values associated with these straits are low or medium, the Sunda Strait route is the most reliable choice. However, under hazardous conditions, shifting to the Lombok Strait route may be more advisable. This underscores the need for the decision makers in shipping companies to conduct thorough risk assessments of maritime nodes and make informed route choices based on prevailing risk values. By navigating varying risk landscapes effectively, tanker carriers can optimize their transportation strategies and ensure the reliable transport of crude oil.

Second, import strategies should be flexible and adaptable, based on evolving risk perceptions and international conditions. As the international situation stabilizes and risks decrease, there may be opportunities to import more crude oil from the Middle East because of the shorter transportation distance. Conversely, if the international situation deteriorates, it may be prudent to diversify imports away from regions with higher risks, such as the Middle East. Increasing imports from more stable regions such as Asia and Latin America can mitigate reliance on routes with inherent risks and potential disruptions. Timely adjustments of import patterns aligned with risk scenarios and geopolitical developments by policymakers can improve the overall transport reliability and supply chain resilience.

Finally, this study has some limitations that should be addressed in future research. The different impacts of various risk factors on maritime nodes are not considered in this study. Therefore, future studies should explore how different risk factors affect the connectivity reliability of nodes and the selection of the most reliable oil shipping route. In addition, we do not consider shipping costs in our study, and a future extension would be to investigate the optimal routes for oil shipments by considering the trade-off between shipping costs and connectivity reliability. Furthermore, it would be valuable to investigate whether the identified shipping routes meet the import requirements in terms of transportation time and capacity reliability. This aspect warrants further study to ensure that the selected routes not only maximize the reliability of connectivity but also fulfill essential transportation criteria.

Data availability statement

Data will be made available on request.

CRediT authorship contribution statement

Shuang Wang: Writing – review & editing, Writing – original draft, Validation, Supervision, Software, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Yan Wang: Writing – review & editing, Writing – original draft, Visualization, Validation, Resources, Methodology, Data curation, Conceptualization. Chengshou Lai: Writing – review & editing, Validation, Formal analysis.

Declaration of competing interest

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

Acknowledgements

This work was supported by the 10.13039/501100001809 National Natural Science Foundation of China [grant Number 72104041 ]; Liaoning Provincial Social Science Foundation of China [grant Number L21CGL009 ]; 10.13039/501100002858 China Postdoctoral Science Foundation [grant Number 2023M740464 ]; and the 10.13039/501100012226 Fundamental Research Funds for the Central Universities [grant Number 3132024178 ].
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