Reliable Liner Shipping Hub Location Problem Considering Hub Failure

Abstract

Hub ports play an important role in reducing the number of container routes and saving the operating costs of liner companies. However, hub failures caused by uncertainties such as natural disasters may cause huge recovery costs. Considering the accidental failure probability of hub ports and the reallocation of feeder ports, a global reliable liner shipping hub location problem (RLSHLP) is studied. We use the k-means algorithm to divide the global shipping network into different communities and propose two mixed integer linear programs to determine the hub port of each community. Then Benders decomposition algorithm is applied to the global RLSHLP and compared with our method. Numerical experiments verify the effectiveness of our method and determine the optimal hub port location scheme considering hub failure.

1. Introduction

The realization of hub and spoke network functions is closely related to hubs. While obtaining economies of scale through transportation between hubs, it is also vulnerable to hub failures. Traditional hub-and-spoke networks ignore the possibility of hub failure when they are designed [1]. However, in practice, operators need to face various situations that cause hub failures and take measures to restore normal network operation in time. Such reactive responses are often costly and inefficient [2]. Improving network reliability is a common method to reduce the impact of hub failure [3,4,5,6]. Therefore, in recent years, many studies have highlighted the importance of considering the influence of hub failure in the network design stage and proposed a series of reliable hub location problems (RHLPs) [7,8,9]. In order to achieve economic advantages and network reliability, the hub locations and conventional routes, as well as the backup hubs and alternative routes in case of interruption, are generally considered when solving the RHLP [10,11]. The research on this problem is helpful for liner companies to make decisions at the strategic level and has certain engineering value in reducing the operating cost of liner companies. For example, in the event of a hub facility failure, the network can be maintained by relying on backup hubs and alternative routes, which does not significantly increase daily operating costs.

In liner shipping, a large number of containers are concentrated at a hub port and transported by large ships deployed between hub ports to obtain economies of scale [12]. Therefore, the location of the hub port is a crucial decision for the operation of liner companies. We call this problem the liner shipping hub location problem (LSHLP). In the traditional hub location problem (HLP), the route planning of cargo is based on the hub-and-spoke structure, which forms the hub-and-spoke network. This hub-and-spoke network is widely used in the field of liner shipping to reduce the operating costs of liner companies [13,14,15].

The current liner shipping industry is not only faced with disruptions caused by natural disasters, terrorist attacks, or strikes but also with the deepening impact of COVID-19, port disruptions become more common [16]. Especially when the hub port that normally operates fails, it may have a significant impact on the shipping network [17]. For a single liner company, hub ports in its hub-and-spoke shipping network are often far away from each other, so that carriers can fully obtain economies of scale of transportation between hubs. However, when the hub port fails, the feeder port may have to be reallocated from the original hub port to other hub ports that are farther away and require higher transportation costs. Therefore, in order to reduce operating costs in case of hub failure, when the interruption occurs, the liner company should reallocate the interrupted routes to the nearest backup hub port as far as possible. From the perspective of a single liner carrier, this paper proposes a global RLSHLP, which considers the impact of hub failure on the global shipping network to minimize transportation costs and then determines the liner shipping network design from the perspective of global optimization.

The main contribution of this paper can be described as the following four parts: (1) By considering hub failure in the LSHLP, this paper introduces backup hubs and alternative routes to investigate the global RLSHLP; (2) For the global RLSHLP, this paper uses the k-means algorithm combined with the elbow method to divide the global shipping network into different communities, and then determines the hub port of each community; (3) Two mixed integer linear programs (MILPs) are proposed for formulating single allocation and multiple allocation RLSHLP; (4) The modified Benders decomposition algorithm is applied to global RLSHLP and compared with our proposed method.

The remainder of the current paper is as follows. In Section 2, we provide a literature review of the RHLP and the LSHLP. Section 3 provides notations and assumptions used in this paper. In Section 4, the global shipping network is divided, and two MILPs are presented. Section 5 provides the Benders decomposition algorithm for global RLSHLP. Section 6 gives numerical experiments, and Section 7 presents our summary.

2. Literature Review

This paper aims to investigate the RLSHLP. In this section, the studies on the RHLP and the LSHLP are summarized.

For HLP, the reliability of paths and the location of hubs play an important role in creating hub networks. We call the combination of the two the RHLP. Kim and O’Kelly (2009) are scholars who studied this kind of problem earlier [18]. Considering the reliability of each link or hub, they put forward the reliable p-hub location problem and applied it to the telecommunication network. Kim (2012) proposed a series of hub location models to alleviate hub failures and ensure that interrupted flow can be rerouted through backup hubs and alternative routes [4]. Zeng et al. (2010) established a reliable single allocation and multiple allocation model considering hub unavailability and alternative routes given the reliability value of each hub [3]. In addition, An et al. (2015), Azizi et al. (2016), Tran et al. (2017), and Rostami et al. (2018) also used backup hubs to model the uncapacitated RHLP to maintain the network operation after hub failures [1,5,6,11]. Mohammadi et al. (2019) proposed a bi-objective capacitated RHLP that takes into account the use of backup hubs and the uncertainty of links [7]. Azizi and Salhi (2021) established a reliable hub location model with multiple capacity levels and flow discount factors by considering backup hubs and the discount factors of links between hubs [9].

Davari et al. (2010), Zarandi et al. (2011), and Mohammadi et al. (2016) studied the RHLP by combining the fuzzy theory [10,19,20]. Parvaresh et al. (2013) and Parvaresh et al. (2014), respectively, designed different bi-level models to study the multiple allocation p-hub location problem [21,22]. The difference is that the upper level of the latter has two objective functions to minimize the normal and worst-case transportation costs. Korani and Eydi (2021) also used the bi-level programming method to study the RHLP and proposed a KKT penalty function solution method [8].

In the field of liner shipping, there are abundant studies on HLP, most of which combine the HLP with the liner shipping network design problem (LSNDP). Sambracos et al. (2004), Fagerholt (2004), and Karlaftis et al. (2009) planned the optimal shipping route based on the location of the hub port and combined it with the fleet deployment problem [23,24,25]. Given hub ports, Imai (2009) analyzed two kinds of liner service networks, hub-and-spoke, and multi-port connection, considering different ship types [26]. Combined with LSNDP and fleet deployment, Gelareh and Pisinger (2011) and Gelareh et al. (2010, 2013) proposed mixed integer models for determining the location of hub ports [27,28,29]. Zheng et al. (2014, 2015) proposed a multi-stage decomposition approach for decomposing the hub-and-spoke LSNDP into hub port location and liner ship route design [30,31]. Based on given hub ports, Bolstad et al. (2019) proposed a path flow formulation to determine the optimal route for shortsea LSNDP [32]. In order to study the relationship between hub location and liner network structure, Chen et al. (2020) established a bi-level programming model considering the shippers’ route choice behavior [33]. Recently, Bütün et al. (2020) proposed a directed cycle hub location and routing problem under congestion to explore a liner network with optimal cost [34].

Sun and Zheng (2016) proposed a two-stage method to measure the hubbing probability of different ports to explore potential hub ports [35]. Zheng et al. (2018) applied the community division method to the liner shipping field, divided the ports in the global network into different communities, and reduced the difficulty of solving the large-scale LSHLP by solving each community separately [13]. Considering the influence of canal effects, Zheng et al. (2019) established several binary linear programming models to explore the LSHLP near the Panama Canal and the Suez Canal [36]. Zheng et al. (2022) proposed that liner shipping has a special spatial structure and explored the impact of the Arctic route on hub location by investigating LSHLP with spatial structure [14]. Corey et al. (2022) studied a kind of the hierarchical hub location problems for a certain region, which provides a reference for the selection of regional hub ports [15].

At present, there are sufficient studies on the RHLP and the LSHLP, but there are few studies on the combination of the two, that is, there are relatively few studies on the location of hub port and shipping network design considering the backup hub port and alternative container routing in liner shipping field. In this paper, a global RLSHLP is proposed to minimize transportation costs under normal and failure conditions and to determine liner shipping networks from a global optimal perspective, considering the influence of hub failure as a disturbing factor.

3. Notations and Assumptions

3.1. Community, Container Routing and Transportation Cost

Generally, community structures are prevalent in reality networks, and the liner shipping network also has this feature [37,38]. The hub port within the community is closely connected with its associated feeder port, and containers between different communities are transported along the main waterway routed via hub ports [13]. Therefore, we can simply regard different hub ports and their related feeder ports as different community. Following Sun et al. (2012), ports in the global liner shipping network can be divided into many different communities [38]. Let $K$ be the community set, $N_k$ be the port set in community $k\in K$, and $H_k$ be the candidate hub port set in community $k\in K$. According to the above characteristics of the liner shipping network, we can assume that at least one hub port should be opened in the community k to provide services for all feeder ports in the community k.

Figure 1 shows the container routing of a port within a community. According to Zheng et al. (2018) and Sun and Zheng (2016), it is assumed that containers are transported between communities along main waterways [13,35]. $0_k^{\ast }$ and ${(|N_k|+1)}_k^{\ast }$, respectively, represent two endpoints of the main waterway connecting community k. Depending on different routes such as the three major trade routes, containers at any port can be transported in different shipping directions. Therefore, for our problem, two container demands of westbound and eastbound of port i are considered (denoted by $Q_i^1$ and $Q_i^2$), respectively, representing the weekly container transport volume associated with these two directions. As shown in Figure 1, for any port i in community k, we consider that $Q_i^1(Q_i^2)$ containers are transported to the endpoint $0_k^{\ast }({(|N_k|+1)}_k^{\ast })$ of the main waterway through hub port j.

For the cost of transporting containers, we assume that it is proportional to the distance of sea transportation. We use $c_{ij}$ to denote the cost of transporting a TEU container between ports (or nodes) i and j, which is calculated as follows:

$$c_{ij}=c_{unit}Di{s}_{ij}$$

(1)

where $Di{s}_{ij}$ represents the distance between ports (or nodes) i and j. According to Zheng et al. (2019), $c_{unit}$ is set at 0.00825 [36]. If feeder port i is assigned to hub port j, ${\omega }_{ij}^1$ and ${\omega }_{ij}^2$ are used to represent the transportation costs in the two directions of westbound and eastbound. Namely, we have

$${\omega }_{ij}^1=(c_{ij}+\alpha c_{j{0}_k^{\ast }})Q_i^1$$

(2)

$${\omega }_{ij}^2=(c_{ij}+\alpha c_{j(|N{|}_k+1)\ast })Q_i^2$$

(3)

where $\alpha$ is the transportation discount factor.

3.2. Hub Failure

In liner shipping, due to severe weather, labor strikes, and other factors, the hub ports that normally operate may be unavailable. As an example, in July and August 2022, the Hamburg Port in Germany, the Liverpool Port, and the Felixstowe Port in the UK were all affected by the strike of dockers, which affected the normal operation of ports. Due to the strike of Felixstowe, liners were forced to divert to other ports, affecting trade worth 800 million [39]. The failure probability of the port is expressed by ${\theta }_i$, and the probability of failure occurring at each hub port is assumed to be independent. The decisions about the location of hub ports and the allocation of feeder ports are made before any failure occurs. When the interruption occurs, the liner company can use the proximity principle to allocate the interrupted route to the nearest backup hub port. We use Figure 2 as an example to describe it in detail. As shown in Figure 2a, normally, containers from ports in Region 1 are transferred through Hub 1 and Hub 2 to Hub 3 in Region 2. When Hub 1 fails and there is no backup hub, containers of Feeders 1–3 need to be transported to a farther Hub 2, and then to Region 2, as shown in Figure 2b. For Feeders 1–3, this change is inconvenient and requires higher transportation costs. Therefore, in our study, Feeder 3 can be selected as a backup hub of Hub 1, as shown in Figure 2c. When Hub1 fails, Feeder 3 takes on the transfer services of Feeders 1 and 2, which can greatly reduce the losses caused by Hub 1 failures.

We also take the community as a unit to explore the impact of a hub failure. Let $R_k$ be the set of port levels for community k. In order to prevent possible failures, each feeder port of community k can theoretically be assigned to $R_k\ge 1$ hub ports at most. After the interruption, each feeder port is served by the nearest normally operating hub port to which it is allocated. If and only if all the allocated hub ports at levels 0, …, r − 1 failed, the allocated hub port at level r for feeder port $i\in N_k$ will serve it. Penalty costs are incurred when all of the allocated hub ports fail.

For any feeder port $i\in N_k$, there is a cost ${\pi }_i$, representing the liner shipping company’s unit penalty cost for not providing service to feeder port i. If ${\pi }_i$ is less than the cost of transporting containers from feeder port i to its allocated hub port j, then this cost may be incurred even if hub port j is operating normally. Therefore, we introduce the emergency hub port for modeling, and let $j=H_k$, failure probability ${\theta }_{H_k}=0$, and transportation cost $c_{i{H}_k}={\pi }_i$.

The feeder port i in community k should be allocated to $R_k$ regular hub ports under optimal conditions, unless i is served by an emergency hub port at certain level $s<R_k$. In addition, since every regular hub port has the possibility of failure, there must be an emergency hub port in community k to provide service for feeder port i. That is, if the feeder port i is assigned to $R_k$ regular hub ports at level 0,…, $R_k-1$, then at level $R_k$, i should be served by emergency hub port $H_k$.

4. Global Shipping Network Division and Hub Port Location

The use of community structure ensures that the optimal hub port of each community can be determined independently, thus greatly reducing the difficulty of solving the HLP considering the global shipping network. For the global RLSHLP, we divide the global shipping network into different communities and then determine the hub location of each community.

4.1. Global Shipping Network Division

In this subsection, the k-means algorithm combined with the elbow method is used to divide the ports in the global shipping network into $K$ different communities [40,41]. The $K$ value is determined by the elbow method based on the sum of squares of errors (SSE). Let $D_i={(d_{i1},d_{i2},\dots ,d_{in})}^{\mathrm{T}}$ be a vector associated with port i. Let $U_k$ be a variable associated with the centroid of community k, which is determined and updated by the k-means algorithm. Then SSE can be defined as follows:

$$SSE={\displaystyle \sum _{k=1}^K{\displaystyle \sum _{i\in N_k}{‖D_i-U_k‖}_2}}$$

(4)

where ${‖\cdot ‖}_2$ represents the 2-norm of the vector.

The process of the k-means algorithm is described as follows:

Step 1:

Set the initial values of vectors $\{D_1,D_2,\dots ,D_n\}$, and parameter $K$.

Step 2:

Randomly select $K$ ports from $\{D_1,D_2,\dots ,D_n\}$ and form $K$ initial centroid vectors $\left\{U_1,U_2,\dots ,U_K\right\}$, where $U_i$ equals the vector ($D_i$) associated with the ith port selected.

Step 3:

Repeat the following three steps to divide the community.

Substep 3.1:

According to $\underset{k\in \{1,2,\dots ,K\}}{\arg \min }{\left|\left|D_i-\left.\left.U_k\right|\right|\right.\right.}_2$, port i is assigned to the nearest community. Update set $N_k$.

Substep 3.2:

Recalculate the vector $U_k$ according to $U_k=\frac{1}{\left|N_k\right|}{\displaystyle \sum _{i\in N_k}{D}_i}$.

Substep 3.3:

If $\left\{U_1,U_2,\dots ,U_K\right\}$ have not changed, go to Step 4, otherwise repeat Step 3.

Step 4:

Output $K$ communities $\left\{N_1,N_2,\dots ,N_K\right\}$.

According to the above method, we can reasonably divide the global shipping network into 10 different communities. As shown in Figure 3, from east to west, there are Northeast Asia, Southeast Asia, Australia and New Zealand, India, and the Arabian Peninsula as well as the East Coast of Africa, the Mediterranean Sea and Black Sea, Europe, West Coast and South Coast of Africa, East Coast of North America and Caribbean Sea, West Coast of North America, and South America. In the following, we explored the RLSHLP of these ten communities.

4.2. Reliable Hub Port Location within the Community

In this subsection, we propose two MILPs to formulate single allocation and multiple allocation RLSHLP to minimize transportation costs under normal and failure conditions, so as to determine the hub port in any community considering hub failure.

4.2.1. Reliable Single Allocation Model

We propose the following decision variables for the single allocation RLSHLP:

$P_{ijr}$: The probability that feeder port i is assigned to hub port j at level r.

$X_{ijr}$: A variable whose value is 1 when feeder port i is assigned to hub port j at level r, and 0 otherwise.

$Z_k$: A variable whose value is 1 when port k is opened as a hub, and 0 otherwise.

The linear programming model of single allocation RLSHLP for community k is as follow:

$$Min{\displaystyle \sum _{i=0}^{N_k-1}{\displaystyle \sum _{j=0}^{H_k}{\displaystyle \sum _{r=0}^{R_k}({\omega }_{ij}^1+{\omega }_{ij}^2)P_{ijr}{X}_{ijr}}}}$$

(5)

s.t.

$${\displaystyle \sum _{j=0}^{H_k}{Z}_j}=p;$$

(6)

$${\displaystyle \sum _{r=0}^{R_k-1}{X}_{ijr}}\le Z_j,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k-1;$$

(7)

$${\displaystyle \sum _{r=0}^{R_k}{X}_{i{H}_{k}r}}=1,\hspace{0.17em}i=0,\dots ,N_k-1;$$

(8)

$${\displaystyle \sum _{j=0}^{H_k}{X}_{ijr}}+{\displaystyle \sum _{s=0}^{r-1}{X}_{i{H}_{k}s}}=1,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}r=0,\dots ,R_k;$$

(9)

$$P_{ij0}=1-{\theta }_j,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k;$$

(10)

$$P_{ijr}=(1-{\theta }_j){\displaystyle \sum _{m=0}^{H_k-1}\frac{{\theta }_m}{1-{\theta }_m}{P}_{i,m,r-1}{X}_{i,m,r-1}},\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=1,\dots ,R_k;$$

(11)

$$Z_j,X_{ijr}\in \{0,1\},\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=0,\dots ,R_k.$$

(12)

The objective function (5) is to minimize the sum of transportation costs under normal and failure conditions. Constraint (6) ensures that p hubs need to be established. Constraints (7) indicate that containers are routed through hub ports. Constraints (8) ensure that any feeder port should be assigned to an emergency hub port at a certain level. Constraints (9) ensure that the feeder port is either allocated to the r-level regular hub port or to the s-level (s < r) or r-level emergency hub port $H_k$. Where, if r = 0, ${\displaystyle \sum _s^{r-1}{X}_{iHs}}=0$. Constraints (10) and (11) describe the probability that hub port j services feeder port i at level r. Constraints (12) give the range of the remaining decision variables.

For the above nonlinear model, we can use a new variable $W_{ijr}$ to replace the nonlinear term $P_{ijr}{X}_{ijr}$ of the product of the continuous variable and the binary variable, so as to linearize the model. As follows, several new constraints have been added to ensure that $W_{ijr}=P_{ijr}{X}_{ijr}$:

$$W_{ijr}\le P_{ijr},\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=0,\dots ,R_k;$$

(13)

$$W_{ijr}\le X_{ijr},\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=0,\dots ,R_k;$$

(14)

$$W_{ijr}\ge 0,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=0,\dots ,R_k;$$

(15)

$$W_{ijr}\ge P_{ijr}+X_{ijr}-1,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=0,\dots ,R_k;$$

(16)

After linearization, the RLSHLP for community k is expressed as follows:

$$Min{\displaystyle \sum _{i=0}^{N_k-1}{\displaystyle \sum _{j=0}^{H_k}{\displaystyle \sum _{r=0}^{R_k}({\omega }_{ij}^1+{\omega }_{ij}^2)W_{ijr}}}}$$

(17)

s.t.

(6)–(10), (13)–(16)

$$P_{ijr}=(1-{\theta }_j){\displaystyle \sum _{m=0}^{H_k-1}\frac{{\theta }_m}{1-{\theta }_m}{W}_{i,m,r-1}},\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=1,\dots ,R_k;$$

(18)

$$Z_j,X_{ijr}\in \{0,1\},\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=0,\dots ,R_k.$$

(19)

4.2.2. Reliable Multiple Allocation Model

We propose the following decision variables for the multiple allocation RLSHLP:

$P_{ijr}^1$: The probability that $Q_i^1$ containers in feeder port i passing through hub port j at level r.

$P_{ijr}^2$: The probability that $Q_i^2$ containers in feeder port i passing through hub port j at level r.

$X_{ijr}^1$: A variable whose value is 1 when $Q_i^1$ containers in feeder port i passing through hub port j at level r, and 0 otherwise.

$X_{ijr}^2$: A variable whose value is 1 when $Q_i^2$ containers in feeder port i passing through hub port j at level r, and 0 otherwise.

$Z_k$: A variable whose value is 1 when port k is opened as a hub, and 0 otherwise.

Similarly, we replace $P_{ijr}^1{X}_{ijr}^1$ and $P_{ijr}^2{X}_{ijr}^2$ with $W_{ijr}^1$ and $W_{ijr}^2$, respectively. The multiple allocation RLSHLP for community k is expressed as follow:

$$Min{\displaystyle \sum _{i=0}^{N_k-1}{\displaystyle \sum _{j=0}^{H_k}{\displaystyle \sum _{r=0}^{R_k}({\omega }_{ij}^1{W}_{ijr}^1+{\omega }_{ij}^2{W}_{ijr}^2)}}}$$

(20)

s.t.

$${\displaystyle \sum _{j=0}^{H_k}{Z}_j}=p;$$

(21)

$${\displaystyle \sum _{r=0}^{R_k-1}{X}_{ijr}^1}\le Z_j,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k-1;$$

(22)

$${\displaystyle \sum _{r=0}^{R_k-1}{X}_{ijr}^2}\le Z_j,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k-1;$$

(23)

$${\displaystyle \sum _{r=0}^{R_k}{X}_{i{H}_{k}r}^1}=1,\hspace{0.17em}i=0,\dots ,N_k-1;$$

(24)

$${\displaystyle \sum _{r=0}^{R_k}{X}_{i{H}_{k}r}^2}=1,\hspace{0.17em}i=0,\dots ,N_k-1;$$

(25)

$${\displaystyle \sum _{j=0}^{H_k}{X}_{ijr}^1}+{\displaystyle \sum _{s=0}^{r-1}{X}_{i{H}_{k}s}^1}=1,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}r=0,\dots ,R_k;$$

(26)

$${\displaystyle \sum _{j=0}^{H_k}{X}_{ijr}^2}+{\displaystyle \sum _{s=0}^{r-1}{X}_{i{H}_{k}s}^2}=1,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}r=0,\dots ,R_k;$$

(27)

$$W_{ijr}^1\le P_{ijr}^1,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=0,\dots ,R_k;$$

(28)

$$W_{ijr}^2\le P_{ijr}^2,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=0,\dots ,R_k;$$

(29)

$$W_{ijr}^1\le X_{ijr}^1,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=0,\dots ,R_k;$$

(30)

$$W_{ijr}^2\le X_{ijr}^2,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=0,\dots ,R_k;$$

(31)

$$W_{ijr}^1\ge 0,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=0,\dots ,R_k;$$

(32)

$$W_{ijr}^2\ge 0,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=0,\dots ,R_k;$$

(33)

$$W_{ijr}^1\ge P_{ijr}^1+X_{ijr}^1-1,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=0,\dots ,R_k;$$

(34)

$$W_{ijr}^2\ge P_{ijr}^2+X_{ijr}^2-1,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=0,\dots ,R_k;$$

(35)

$$P_{ij0}^1=1-{\theta }_j,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k;$$

(36)

$$P_{ij0}^2=1-{\theta }_j,\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k;$$

(37)

$$P_{ijr}^1=(1-{\theta }_j){\displaystyle \sum _{k=0}^{H_k-1}\frac{{\theta }_k}{1-{\theta }_k}{W}_{i,k,r-1}^1},\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=1,\dots ,R_k;$$

(38)

$$P_{ijr}^2=(1-{\theta }_j){\displaystyle \sum _{k=0}^{H-1}\frac{{\theta }_k}{1-{\theta }_k}{W}_{i,k,r-1}^2},\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=1,\dots ,R_k;$$

(39)

$$Z_j,X_{ijr}^1,X_{ijr}^2\in \{0,1\},\hspace{0.17em}i=0,\dots ,N_k-1,\hspace{0.17em}j=0,\dots ,H_k,\hspace{0.17em}r=0,\dots ,R_k.$$

(40)

The objective function (20) is to minimize the sum of transportation costs under normal and failure conditions. Constraint (21) ensures that p hubs need to be established. Constraints (22) and (23) indicate that containers are routed through hub ports in both east and west directions. Constraints (24) and (25) ensure that any feeder port should be assigned to a certain level of emergency hub port in both east and west directions. Constraints (26) and (27) have the same meaning as constraint (9). Constraints (28)–(35) are similar to constraints (13)–(16). Constraints (36)–(39) describe the probability that hub port j services feeder port i at level r in both east and west directions.

5. Benders Decomposition for the Global RLSHLP

For the global RLSHLP, solving each community separately can greatly reduce the difficulty and complexity of the solution, so Cplex can completely meet the requirements. To highlight the advantages of our proposed method, we use a modified Benders decomposition algorithm to solve the global RLSHLP and compare it with our method.

The global RLSHLP is formulated as:

$$Min{\displaystyle \sum _{i=0}^{N-1}{\displaystyle \sum _{j=0}^H{\displaystyle \sum _{r=0}^R{c}_{ij}{Q}_i{W}_{ijr}}}}$$

(41)

s.t.

$${\displaystyle \sum _{j=0}^H{Z}_j}=p;$$

(42)

$${\displaystyle \sum _{r=0}^{R-1}{X}_{ijr}}\le Z_j,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H-1;$$

(43)

$${\displaystyle \sum _{r=0}^R{X}_{iHr}}=1,\hspace{0.17em}i=0,\dots ,N-1;$$

(44)

$${\displaystyle \sum _{j=0}^H{X}_{ijr}}+{\displaystyle \sum _{s=0}^{r-1}{X}_{iHs}}=1,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}r=0,\dots ,R;$$

(45)

$$W_{ijr}\le P_{ijr},\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=0,\dots ,R;$$

(46)

$$W_{ijr}\le X_{ijr},\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=0,\dots ,R;$$

(47)

$$W_{ijr}\ge P_{ijr}+X_{ijr}-1,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=0,\dots ,R;$$

(48)

$$W_{ijr}\ge 0,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=0,\dots ,R;$$

(49)

$$P_{ij0}=1-{\theta }_j,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H;$$

(50)

$$P_{ijr}=(1-{\theta }_j){\displaystyle \sum _{k=0}^{H-1}\frac{{\theta }_k}{1-{\theta }_k}{W}_{i,k,r-1}},\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=1,\dots ,R;$$

(51)

$$Z_j,X_{ijr}\in \{0,1\},\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=0,\dots ,R.$$

(52)

where $N$ and $H$ are the set of ports and the set of candidate hub ports, respectively, and $R$ is the set of port levels. $Q_i$ is used to denote the container demand of port i. The objective function (41) is to minimize the sum of transportation costs under normal and failure conditions. Constraints (42)–(45) have the same meaning as Constraints (6)–(9). Constraints (46)–(49) are similar to constraints (13)–(16). Constraints (50) and (51) describe the probability that hub port j services feeder port i at level r.

By fixing integer variable $Z_j=Z_j^t$ in the iterative process, the following linear subproblem (SP) is obtained:

$$Min{\displaystyle \sum _{i=0}^{N-1}{\displaystyle \sum _{j=0}^H{\displaystyle \sum _{r=0}^R{c}_{ij}{Q}_i{W}_{ijr}}}}$$

(53)

s.t.

$${\displaystyle \sum _{r=0}^{R-1}{X}_{ijr}}\le Z_j^t,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H-1;$$

(54)

$${\displaystyle \sum _{r=0}^R{X}_{iHr}}=1,\hspace{0.17em}i=0,\dots ,N-1;$$

(55)

$${\displaystyle \sum _{j=0}^H{X}_{ijr}}+{\displaystyle \sum _{s=0}^{r-1}{X}_{iHs}}=1,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}r=0,\dots ,R;$$

(56)

$$W_{ijr}\le P_{ijr},\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=0,\dots ,R;$$

(57)

$$W_{ijr}\le X_{ijr},\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=0,\dots ,R;$$

(58)

$$W_{ijr}\ge P_{ijr}+X_{ijr}-1,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=0,\dots ,R;$$

(59)

$$P_{ij0}=1-{\theta }_j,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H;$$

(60)

$$P_{ijr}=(1-{\theta }_j){\displaystyle \sum _{k=0}^{H-1}\frac{{\theta }_k}{1-{\theta }_k}{W}_{i,k,r-1}},\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=1,\dots ,R;$$

(61)

$$X_{ijr}\in \{0,1\},\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=0,\dots ,R;$$

(62)

$$W_{ijr}\ge 0,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=0,\dots ,R.$$

(63)

After associating the dual variables $(a_{ij},b_i,c_{ir},d_{ijr},e_{ijr},f_{ijr},g_{ij},h_{ijr})$ to Constraints (54)–(61), respectively, the dual SP can be written as follows:

$$M\mathrm{ax}{\displaystyle \sum _i^{N-1}{b}_i}-{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^{H-1}{a}_{ij}{Z}_j^t}}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _r^R{c}_{ir}}}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^H{g}_{ij}(1-{\theta }_j)}}-{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^H{\displaystyle \sum _r^R{f}_{ijr}}}}$$

(64)

s.t.

$$\begin{array}{l}-d_{ijr}-e_{ijr}+f_{ijr}-(1-{\theta }_j)\frac{{\theta }_k}{1-{\theta }_k}{h}_{i,k,r-1}\le c_{ij}{Q}_i,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}\\ j=0,\dots ,H,\hspace{0.17em}k=0,\dots ,H-1,\hspace{0.17em}r=1,\dots ,R;\end{array}$$

(65)

$$-d_{ijr}-e_{ijr}+f_{ijr}\le c_{ij}{Q}_i,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=0;$$

(66)

$$e_{ijr}-f_{ijr}+c_{ir}\le 0,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H-1,\hspace{0.17em}r=R;$$

(67)

$$-a_{ij}+e_{ijr}-f_{ijr}+c_{ir}\le 0,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H-1,\hspace{0.17em}r=0,\dots ,R-1;$$

(68)

$$b_i+e_{ijr}-f_{ijr}+c_{ir}\le 0,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=H,\hspace{0.17em}r=0;$$

(69)

$$b_i+e_{ijr}-f_{ijr}+c_{ir}+c_{is}\le 0,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=H,\hspace{0.17em}r=1,\dots ,R,\hspace{0.17em}s=0,\dots ,r-1;$$

(70)

$$d_{ijr}-f_{ijr}+h_{ijr}\le 0,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=1,\dots ,R;$$

(71)

$$g_{ij}+d_{ijr}-f_{ijr}\le 0,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=0;$$

(72)

$$a_{ij}\ge 0,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H-1;$$

(73)

$$d_{ijr},e_{ijr},f_{ijr}\ge 0,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=0,\dots ,R;$$

(74)

$$b_i,c_{ir},g_{ij}\in ℝ,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=0,\dots ,R;$$

(75)

$$h_{ijr}\in ℝ,\hspace{0.17em}i=0,\dots ,N-1,\hspace{0.17em}j=0,\dots ,H,\hspace{0.17em}r=1,\dots ,R.$$

(76)

According to the dual objective function (64), the following constraint can be obtained in the iterative process:

$$\eta +{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^{H-1}{a}_{ij}^t{Z}_j}}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^H{\displaystyle \sum _r^R{f}_{ijr}^t}}}\ge {\displaystyle \sum _i^{N-1}{b}_i^t}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _r^R{c}_{ir}^t}}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^H{g}_{ij}^t(1-{\theta }_j)}}$$

(77)

where, t is the current iteration number, $(a_{ij}^t,b_i^t,c_{ir}^t,d_{ijr}^t,e_{ijr}^t,f_{ijr}^t,g_{ij}^t,h_{ijr}^t)$ are the optimal solution of the dual variable obtained by calculating the dual SP at iteration t, and $\eta$ is the total transportation cost. Therefore, we can obtain the following master problem (MP):

$$Min \eta$$

(78)

s.t.

$$\eta +{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^{H-1}{a}_{ij}^t{Z}_j}}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^H{\displaystyle \sum _r^R{f}_{ijr}^t}}}\ge {\displaystyle \sum _i^{N-1}{b}_i^t}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _r^R{c}_{ir}^t}}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^H{g}_{ij}^t(1-{\theta }_j)}};$$

(79)

$${\displaystyle \sum _{j=0}^H{Z}_j}=p;$$

(80)

$$\eta \ge 0;$$

(81)

$$Z_j\in \{0,1\},\hspace{0.17em}j=0,\dots ,H.$$

(82)

We use $UB$ as the upper bound and $LB$ as the lower bound. $v_{MP}^{\ast }$ and $v_{SP}^{\ast }$ represent the current optimal solution value of the MP and SP, respectively. The classical Benders decomposition algorithm is shown in Algorithm 1.

Algorithm 1: Classical Benders Algorithm

$UB\leftarrow +\infty$, $LB\leftarrow 0$, $t\leftarrow 0$ while ($UB>LB$) do   Solve the MP (78)–(82) to obtain $v_{MP}^{*}$ and $Z_j$   Update $LB\leftarrow v_{MP}^{*}$, $Z_j^t\leftarrow Z_j$   Solve the SP (64)–(76) to obtain $(a_{ij}^t,b_i^t,c_{ir}^t,d_{ijr}^t,e_{ijr}^t,f_{ijr}^t,g_{ij}^t,h_{ijr}^t)$   Add a new Benders cut constraint (77) in the MP (78)–(82)   if ($v_{DS}^{*}<UB$) then    $UB\leftarrow v_{DS}^{*}$   end if   $t\leftarrow t+1$ end while

In the above Benders decomposition algorithm, the solution of MP is a lower bound of the objective value of the original problem (41)–(52). When the lower bound of the objective function gradually approaches its upper bound, the optimal solution to the original problem can be found. According to Geoffrion and Graves (1974) and R.S. de Camargo et al. (2008), the MP is only used to obtain a feasible solution, and it can be terminated when the feasible solution generated by the MP is better than the current optimal solution [42,43]. Therefore, an allowed error margin $\epsilon >0$ is introduced to improve the MP, and the algorithm terminates when the MP cannot find a better feasible solution better than $UB-\epsilon$. In this case, the optimal feasible solution is the $\epsilon$-solution of the original problem (41)–(52).

By modifying the objective function (78) of the MP and replacing $\eta$, we have

$${\displaystyle \sum _i^{N-1}{b}_i^t}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _r^R{c}_{ir}^t}}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^H{g}_{ij}^t(1-{\theta }_j)}}-{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^{H-1}{a}_{ij}^t{Z}_j}}-{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^H{\displaystyle \sum _r^R{f}_{ijr}^t}}}$$

(83)

In order to obtain a better solution than the existing upper bound, thus forming a new Benders cut:

$${\displaystyle \sum _i^{N-1}{b}_i^t}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _r^R{c}_{ir}^t}}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^H{g}_{ij}^t(1-{\theta }_j)}}-{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^{H-1}{a}_{ij}^t{Z}_j}}-{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^H{\displaystyle \sum _r^R{f}_{ijr}^t}}}\le UB-\epsilon$$

(84)

The MP can be replaced as follows:

$$Min{\displaystyle \sum _i^{N-1}{b}_i^t}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _r^R{c}_{ir}^t}}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^H{g}_{ij}^t(1-{\theta }_j)}}-{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^{H-1}{a}_{ij}^t{Z}_j}}-{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^H{\displaystyle \sum _r^R{f}_{ijr}^t}}}$$

(85)

s.t.

$${\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^H{\displaystyle \sum _r^R{f}_{ijr}^t}}}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^{H-1}{a}_{ij}^t{Z}_j}}\ge {\displaystyle \sum _i^{N-1}{b}_i^t}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _r^R{c}_{ir}^t}}+{\displaystyle \sum _i^{N-1}{\displaystyle \sum _j^H{g}_{ij}^t(1-{\theta }_j)}}-(UB-\epsilon )$$

(86)

$${\displaystyle \sum _{j=0}^H{Z}_j}=p;$$

(87)

$$Z_j\in \{0,1\},\hspace{0.17em}j=0,\dots ,H.$$

(88)

Following Geoffrion and Graves (1974) and R.S. de Camargo et al. (2008), the modified Benders decomposition algorithm is shown in Algorithm 2 [42,43].

Algorithm 2: Modified Benders Algorithm

$UB\leftarrow +\infty$, $t\leftarrow 0$, terminate←false while (terminate = false) do   Solve the MP (85)–(88)   if (solution is infeasible) then    terminate←ture   else    Obtain $Z_j^t$   Update $Z_j\leftarrow Z_j^t$   Solve the SP (64)–(76) to obtain $(a_{ij}^t,b_i^t,c_{ir}^t,d_{ijr}^t,e_{ijr}^t,f_{ijr}^t,g_{ij}^t,h_{ijr}^t)$   Add a new Benders cut constraint (84) in MP (85)–(88)   Update $UB\leftarrow U{B}^t$   end if   $t\leftarrow t+1$ end while

6. Numerical Experiments

In this section, a global shipping network consisting of 188 ports for container transportation is used to verify our solution method, as shown in Figure 3, where the container demand of all ports is provided by Maersk. As shown above, CPLEX can effectively solve our proposed models for a single community, where CPLEX runs on a 3.7 GHz Dual-Core PC with 8 GB of RAM. When running CPLEX, the gap is set to 0.001.

6.1. Computational Results of Different Methods

In this section, we take the single allocation RLSLHP as an example to show the comparison results between our solution method of hub port location for each community individually and the overall solution using the Benders algorithm. The parameter $\epsilon$ is set to 0.01, and the failure probability ${\theta }_i$ is randomly generated between 0.01 and 0.05.

Table 1. Computational results of different methods.

$\mathit{\alpha }$	R	CPLEX	CBD		MBD
		CPU Time(s)	Iteration	CPU Time(s)	Iteration	CPU Time(s)
0.4	2	2.74	22	1881.55	7	422.91
0.6	2	2.77	20	1584.61	7	419.12
0.8	2	2.65	26	2463.86	7	420.71
0.4	3	3.97	27	2478.82	8	464.60
0.6	3	3.96	27	2501.41	7	490.42
0.8	3	3.91	32	2988.67	7	414.01

shows the calculated results of the tests we performed.

By dividing the global liner shipping network, the hub location results are solved separately for each community. At this time, all ports in each community have the possibility of becoming hub ports. Due to the reduction of solving difficulty and complexity, this method can be solved within 4 s. When using the Benders decomposition algorithm for the overall solution, due to the large data set, we select 20 ports with large container demand in the global shipping network as candidate hub ports. Whether classical or modified Benders algorithm, the solution time is much longer than our method.

6.2. Hub Location Results

6.2.1. Single Allocation

Here we mainly show the hub location results for solving the single allocation RLSHLP by introducing the hubbing probability of the port. The hubbing probability of any port is measured by its proportion opened as a hub, which is obtained by solving the single allocation RLSHLP when different values of $\alpha$ and $p$ are considered. Since any feeder port must be assigned to an emergency hub port of a certain level, we generally set $R=p+1$. Figure 4 shows the hub location results with different values of $p$, when $\alpha$ is changed from 0.1 to 0.9.

As can be seen from Figure 4, Los Angeles, Rotterdam, Gioia Tauro, Jawaharlal Nehru, and Shanghai must be the hub port (i.e., their hubbing probability is 1) no matter what values $\alpha$ and $p$ are taken. These ports are almost all located in the central area of their respective communities and have some of the largest demand for containers within their community. In addition, Manzanillo and Singapore also have great potential as hub ports for different values of $\alpha$ and $p$. Most of the ports shown in Figure 4 are also well-known hub ports in the world, which proves the rationality of our proposed method to a certain extent. It should be noted that Australia and New Zealand form a small community of seven ports due to their geographical location. There can only be one hub port in the region at most, so the potential of Auckland cannot be compared with the above ports.

Based on Figure 4, we find that when $\alpha$ is small, Miami is more suitable than Manzanillo to become a hub port for communities on the East Coast of North America and the Caribbean Sea. Similarly, for Southeast Asia, Laem Chabang is superior to Saigon, and Tanjung Pelepas is superior to Singapore. For the community around India, the Arabian Peninsula, and the East Coast of Africa, Jebel Ali is superior to Jeddah. For the community in Northeast Asia, Busan has more advantages than Yokohama. For the community associated with Australia and New Zealand, Brisbane and Auckland are superior to Sydney. This is closely related to the container demand of ports, that is, when $\alpha$ is small, ports with large container demand have a higher probability of becoming hub ports.

6.2.2. Multiple Allocation

Here, we mainly show the hub location results for solving multiple allocation RLSHLP, by changing $\alpha$ from 0.1 to 0.9, as shown in Figure 5. Two and three hub ports are opened for each community in Figure 5a,b, respectively. Compared with Figure 4, most of the ports that can be selected as hubs in the single allocation scheme also have strong advantages in the multiple allocation scheme. It can be seen that the global RLSHLP can identify the most important hub port in reality, but its location results are not sensitive to the change in allocation scheme. For some communities (e.g., East Coast of North America and Caribbean Sea, Southeast Asia, and West Coast of North America), different allocation schemes do not change the hubbing probability of ports within the community, that is, the change in allocation schemes has no effect on the hub location results of these communities. For some communities (such as Europe), Bremerhaven is chosen as the hub port, rather than some more well-known ports such as Antwerp and Hamburg. This is because some famous ports in the world to become hub ports not only need to have superior geographical location, but also related to their own large container demand. The difference is that as $\alpha$ gets larger, for some communities (such as Northeast Asia), the choice of hub port is gradually closer to the community boundary.

6.3. Impact of Hub Failure

In this subsection, we take Northeast Asia as an example to show the differences between RLSHLP and the classical hub location problem (CHLP) under a single allocation scheme, when $p=3$, $\alpha =0.6$, and the failure probability of ports in the community are shown in

Table 2. Failure probability of each port in the RLSHLP.

No.	City	$\mathit{\theta }$	No.	City	$\mathit{\theta }$	No.	City	$\mathit{\theta }$
0	Guam	0.034	7	Fuzhou	0.026	14	Busan	0.026
1	Manila	0.049	8	Shanghai	0.032	15	Nagoya	0.031
2	Hong Kong	0.025	9	Hakata	0.043	16	Yokohama	0.042
3	Yantian	0.030	10	Osaka	0.015	17	Tokyo	0.026
4	Kaohsiung	0.021	11	Kobe	0.039	18	Qingdao	0.049
5	Xiamen	0.013	12	Lianyungang	0.018	19	Dalian	0.042
6	Keelung	0.020	13	Shimizu	0.037

. There are many ports in Northeast Asia and their distribution is relatively wide. As the main container export area, Northeast Asia occupies an important position in the global container shipping network. As shown in Figure 6, in the CHLP, Yokohama is selected as one of the hub ports in Northeast Asia, and it is replaced by Busan in consideration of hub failure. The feeder allocation is also changed accordingly. In addition to Yokohama and the feeder ports originally served by Yokohama being completely assigned to Busan, Dalian is also reassigned to Busan.

The objective function of the RLSHLP model includes the costs under both normal and failure conditions, while the objective function of the CHLP model only calculates the costs under normal conditions. The two cannot be directly compared, so we measure the service quality of liner companies by comparing the total container traffic volumes of the CHLP and the RLSHLP. For the case we discussed (i.e., $p=3$, $\alpha =0.6$), the total number of containers transported in the CHLP is 146,738 and 149,404 in the RLSHLP, an increase of 1.8%. Then we also calculate the European community. We find that the CHLP and the RLSHLP have the same results of hub location and feeder allocation, unlike the Northeast Asia community, which can clearly see the difference in hub location results. For the CHLP and the RLSHLP, Rotterdam, Bremerhaven, and Gothenburg are all selected as hub ports. However, the design of the liner shipping network considering hub failure also plays a role in transporting more containers. The total number of containers transported in the CHLP is 54,603 and 55,687 in the RLSHLP, an increase of 2%. It can be seen that the liner shipping network designed considering the disturbance factors of hub failure has better performance.

For Northeast Asia, we collect ports with major accidents since 2020, including Busan, Manila, Hakata, Yantian, and Kaohsiung. According to actual cases, we raise the failure probability of these ports to 0.2 based on Table 2, so as to make the calculation more realistic. We find that the total number of containers transported in the CHLP decreased to 137,545. In the RLSHLP, Busan is replaced as a hub by Yokohama due to high failure probability, and the number of containers transported increased by 9% compared with the CHLP.

7. Summary

This paper studies the global reliable liner shipping hub location problem (RLSHLP) under single and multiple allocation schemes. By dividing the global shipping network into different communities and considering hub failure, two mixed integer linear programs are proposed to find a reliable and cost-effective liner shipping hub port location scheme for each community. In order to highlight the advantages of our solution method, we use a modified Benders decomposition algorithm to solve the global RLSHLP and compare it with our method.

Numerical results show that (i) our method greatly reduces the solution time of the global RLSHLP, and the most well-known hub ports in the world can be found, which proves the effectiveness of our method; (ii) For most famous ports, different allocation schemes have no significant impact on their hub status; (iii) For the selected hub port, on the one hand, they occupy a superior geographical location (such as the central area of the community), on the other hand, they have a large demand for containers; (iv) Compared with the CHLP, the liner shipping network composed of hub location and feeder allocation obtained by solving the RLSHLP has better performance, which can greatly improve the container transportation volume.