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Empty container exchange among liner carriers
Transportation Research Part E 83 (2015) 158–169
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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre
Empty container exchange among liner carriers Jianfeng Zheng a,⇑, Zhuo Sun a, Ziyou Gao b a b
Transportation Management College, Dalian Maritime University, Dalian 116026, China School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China
a r t i c l e
i n f o
Article history: Received 22 June 2015 Received in revised form 21 September 2015 Accepted 22 September 2015
Keywords: Empty container allocation Value of empty container Two-stage optimization method Inverse optimization
a b s t r a c t In an attempt to reduce the empty container repositioning costs, this paper studies an empty container allocation problem considering the coordination among liner carriers. We further measure the perceived values of empty container at different ports. The perceived values of empty container at the surplus (deficit) ports are described by the profits (empty container exchange costs paid) for delivering empty containers. To solve our problems, we propose a two-stage optimization method. In stage I, liner carriers are guided to pursue a centralized optimization solution of empty container allocation for all related liner carriers. In stage II, the inverse optimization technique is used to determine the empty container exchange costs, which are paid to liner carriers for exchanging empty containers and following the centralized optimization solution. The profits at the surplus ports are calculated with respect to the empty container exchange costs at the deficit ports. Finally, numerical experiments on an Asia–Europe–Oceania shipping service network are discussed. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction In the liner shipping industry, empty container repositioning is a challenge for liner carriers due to the high costs. Since 1993, empty container movements have constituted about 20% of the total ocean container movements (Song and Dong, 2011). In 2003, the repositioning cost was up to $11 billion (Bonney, 2004), and that in 2010 was about $23.4 billion (Drewry, 2011; Tran and Haasis, 2015). Song et al. (2005) estimated that the repositioning cost accounts for 27% of the total world fleet running cost. Because of the trade imbalances between the major trading regions, empty container movements cannot be avoided completely. However, minimizing these costly activities would considerably reduce the operating costs of liner carriers. In an attempt to reduce the cost on repositioning empty containers, this paper proposes an empty container allocation problem considering the coordination among liner carriers, where empty containers of any single liner carrier can be repositioned to serve the needs of other liner carriers. Hence, empty containers are exchanged among liner carriers. Similar to slot exchange agreements between liner carriers in a strategic alliance, empty container exchange agreements can be signed between liner carriers to reduce the repositioning costs of liner carriers. Generally, empty container movements do not generate revenue for liner carriers. In order to motivate liner carriers to follow a centralized optimization solution of empty container allocation for all related liner carriers, the extra incentives should be introduced to guide liner carriers. Generally, these incentives can be described by the profits (costs paid) for delivering empty containers from (to)
⇑ Corresponding author. E-mail address: [email protected] (J. Zheng). http://dx.doi.org/10.1016/j.tre.2015.09.007 1366-5545/Ó 2015 Elsevier Ltd. All rights reserved.
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the surplus (deficit) ports, which have a surplus (deficit) of empty containers. This motivates us to measure the perceived values of empty container at different ports. There have been many studies related to the empty container allocation problem or the empty container repositioning issue. Crainic et al. (1993) developed two dynamic deterministic formulations and a stochastic formulation for empty container allocation in a land distribution and transportation system. Cheung and Chen (1998) investigated the dynamic empty container allocation problem, which is formulated as a two-stage stochastic network model. A stochastic quasigradient method and a stochastic hybrid approximation procedure were applied to solve the problem. Erera et al. (2009) developed a robust optimization framework for dynamic empty repositioning problems modeled using time–space networks. Li et al. (2004) discussed empty container management at a port and derived the optimal pairs of critical policies, (U, D) for this port. Namely, if the number of empty containers at this port is less than U, empty containers are imported up to U, or empty containers are exported down to D if the number of empty containers at this port is larger than D. Later, Li et al. (2007) extended this problem by considering multiple ports. Jula et al. (2006) studied empty container movements by optimizing the empty container reuse. The dynamic empty container reuse was modeled and optimization techniques were developed to optimize the empty container operations. Lam et al. (2007) presented an approximate dynamic programming approach, in order to obtain the effective empty container relocation strategies. Feng and Chang (2008) investigated the repositioning of empty containers for intra-Asia liner shipping. Song and Carter (2009) studied general empty container balancing strategies depending on whether shipping lines are coordinating the container flows over different routes and whether they are willing to share container fleets. Moon et al. (2010) proposed an empty container repositioning problem considering leasing and purchasing. To address this problem, they presented a mixed-integer linear optimization model and developed a genetic algorithm to solve it. Shintani et al. (2010) analyzed the possibility to save the container fleet management costs in repositioning empty containers by using foldable containers. Later, Moon et al. (2013) compared the foldable containers with the standard containers on the cost for repositioning empty containers. Numerical experiments demonstrated the economic feasibility of foldable containers. Di Francesco et al. (2009) addressed an empty container repositioning problem under uncertainty, where the historical data were inappropriate for estimating uncertain parameters. In order to solve this problem, a time-extended multi-scenario optimization model was developed. Later, Di Francesco et al. (2013) studied an empty container repositioning problem under uncertain port disruptions. Long et al. (2012, 2015) investigated an empty container repositioning problem with uncertainties, by using a sample average approximation method. Song and Dong (2011) discussed an empty container repositioning policy with flexible destination ports. Bell et al. (2011, 2013) proposed two types of container assignment models (i.e., a frequency-based container assignment model and a cost-based container assignment model), in which both laden containers and empty containers were considered. Recently, Wang et al. (2015) extended these two container assignment models by proposing several profit-based container assignment models. In addition, readers can refer to the references in two reviews on empty container repositioning (Braekers et al., 2011; Song and Dong, 2015). Furthermore, some researchers explored the combined optimization problems in liner shipping with empty container repositioning. Shintani et al. (2007), Meng and Wang (2011), Song and Dong (2013) investigated the liner shipping network design problem, as well as considering the repositioning of empty containers. Dong and Song (2009) addressed the joint problem of container fleet sizing and empty container repositioning. Brouer et al. (2011) studied the cargo allocation problem with the repositioning of empty containers. Song and Dong (2012) investigated cargo routing, together with empty container repositioning. For more optimization problems related to ship routing and scheduling in liner shipping, please refer to some review papers (Christiansen et al., 2004, 2013; Meng et al., 2014). To the best of our knowledge, there is currently no optimization model on the empty container allocation problem considering the coordination among liner carriers. Le (2003) studied a related problem based on a neutral Internet-based information exchange platform, which may facilitate empty container reuse and sharing empty containers among liner carriers. Theofanis and Boile (2009) mentioned that empty containers of a liner carrier can be used to match the needs of other liner carriers. Both of these works discussed the empty container repositioning strategies from a qualitative point of view. Obviously, the coordination among liner carriers increases the flexibility of an empty container repositioning system by exchanging empty containers among liner carriers and offers an opportunity to reduce the repositioning costs of liner carriers. However, the coordination formation among liner carriers is challenging. Generally, the goal of a liner carrier is the maximization of its own profit (or minimization of its own cost). Furthermore, some liner carriers may collude to obtain a larger profit (or a lower cost), as compared with the coordination among more liner carriers. Hence, the coordination stability among liner carriers is also a challenge. This paper aims to resolve these problems in the study of empty container allocation considering the coordination among liner carriers. Furthermore, we will measure the perceived values of empty container at different ports, under the coordination among liner carriers. The rest of this paper is organized as follows. Notation, assumptions and problem description are described in Section 2. A two-stage optimization method is presented in Section 3. Numerical results are given in Section 4. The conclusions are shown in Section 5. 2. Notation, assumptions and problem description Some mathematical notations have to be defined in order to facilitate description and formulation of the problem.
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Parameters L Set of liner carriers P Set of ports S Set of surplus ports Sk Set of surplus ports for liner carrier k 2 L W Set of deficit ports Wk Set of deficit ports for liner carrier k 2 L Qk Set of OD (Origin-Destination) demands of liner carrier k 2 L qkij Weekly number of containers of liner carrier k transported from origin port i to destination port j, i.e., the OD demand kij The average transportation cost for delivering one empty container from port i to port j Decision variables xkm Weekly number of empty containers transported from surplus port i of liner carrier k 2 L to deficit port j of liner ij carrier m 2 L costj The empty container exchange cost for providing one TEU (Twenty-foot Equivalent Unit) empty container at deficit port j 2 W
Generally, shippers will choose the proper liner carriers for delivering their containers, and they will sign a short-term or long-term contract with their chosen liner carriers (Zheng et al., 2015). As described in Christiansen et al. (2007), it is not unusual that some liner carriers do between 80% and 95% of their business under the long-term contracts, most of which are negotiated once a year between liner carriers and shippers. According to the fixed container demands of different liner carriers, the number of empty containers accumulated at the surplus ports or lacked at the deficit ports for different liner carriers can be obtained. Generally, a practical ship route (i.e., port calling sequence by a ship) often maintains a weekly service frequency, and the weekly container demands of different liner carriers can be used to determine the distribution of empty containers at different ports. In order to simplify our problem, we assume that the weekly container demands of different liner carriers are fixed and given in advance. Given the weekly OD demand from origin port i to destination port j of liner carrier k, qkij ð8qkij 2 Qk Þ, let nki denote the difference between the incoming and outgoing container flow for liner carrier k at port i, calculated as follows:
nki ¼
X j2P
qkji
X qkij ;
8 k 2 L; 8i 2 P
ð1Þ
j2P
Clearly, nki represents the empty container supply or demand of liner carrier k at port i. When nki takes a positive value, port i is a surplus port of liner carrier k, and port i is a deficit port of liner carrier k, when its value is negative. Generally, the empty container allocating processes are typically time consuming and depend on the space available on vessels, both of which are not considered in this paper. Namely, we assume: (i) travel times for delivering empty containers are neglected; (ii) each liner carrier has sufficient space for repositioning empty containers. For simplicity, this paper mainly considers the average transportation cost for delivering empty containers from the surplus ports to the deficit ports. Note that the empty container repositioning is constrained by vessel spare capacities since laden containers have priority to be shipped (Dong and Song, 2009). In practice, empty containers may be transported by a less cost-efficient liner carrier due to capacity constraints. In other words, the transportation cost for transporting empty containers from some certain surplus ports to some certain deficit port may be various for different liner carriers, especially when considering the coordination among liner carriers. For instance, one liner carrier can buy some vessel slots from other liner carriers, leading to the different costs on transporting (laden and empty) containers between some certain ports for different liner carriers. In this paper, we assume that the average transportation cost for transporting one empty container from any surplus port to any deficit port is identical for different liner carriers. As mentioned before, one aim of this paper is to measure the perceived values of empty container at different ports, which are defined as follows. For a particular surplus port, its perceived value of empty container is described by the unit profit for delivering one empty container from this surplus port to any possible deficit port. For a particular deficit port, its perceived value of empty container is represented by the cost paid to liner carriers for delivering one empty container to this deficit port. This cost is termed ‘‘the empty container exchange cost” in this paper. Since empty container movements do not generate revenue for liner carriers, we assume that the unit profits for delivering empty containers from the surplus ports only depend on the average transportation cost for transporting empty containers and the empty container exchanges costs at the deficit ports, independent of any other aspects (e.g., container leasing prices). Because of the coordination among liner carriers, we assume that empty containers are repositioned in such a way that the total transportation cost on delivering empty containers for all related liner carriers is minimized, leading to a centralized optimization solution of empty container allocation. Then, we should determine the cost allocation (or profit sharing) among liner carriers. For a small number of participating liner carriers, using Shapley value (Shapley, 1953) would be a meaningful approach. Furthermore, this paper will measure the perceived values of empty container at different ports, based on the centralized optimization solution. To solve our problems, we propose a two-stage optimization method, as shown in Section 3.
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3. Two-stage optimization method Here, we propose a two-stage optimization method for solving our problems. In stage I, a centralized optimization solution of empty container allocation for all related liner carriers can be obtained, as shown in Section 3.1. In stage II, we aim to determine the perceived values of empty container at different ports based on the solution obtained in stage I, as shown in Section 3.2. 3.1. Stage I In stage I, we aim to determine the weekly number of empty containers delivered from surplus port i of liner carrier k 2 L to deficit port j of liner carrier m 2 L (i.e., xkm ij ) to obtain a centralized optimization solution of empty container allocation for all related liner carriers. The mathematical programming model (denoted by P) to pursue this centralized optimization solution can be formulated as follows:
XXX X
min
ðkij xkm ij Þ
ð2Þ
8 i 2 Sk ; 8 k 2 L;
ð3Þ
k2L m2L i2Sk j2Wm
subject to
XX
k xkm ij ¼ ni ;
m2L j2Wm
XX m xkm ij ¼ nj ;
8 j 2 Wm ; 8 m 2 L;
ð4Þ
k2L i2Sk
xkm ij P 0;
8 i 2 Sk ; 8 j 2 Wm ; 8 k; m 2 L:
ð5Þ
The objective function (2) minimizes the total transportation cost for delivering empty containers of all related liner carriers from the surplus ports to the deficit ports. Constraints (3) and (4) ensure that empty containers should be balanced after as a nonnegative variable. Note that the above formulation also allocating empty containers. Constraints (5) denote xkm ij accounts for the case k = m, where any single liner carrier delivers its own empty containers from its surplus ports to its deficit ports without considering the coordination among liner carriers. 3.2. Stage II Based on the optimization model P ((2–5)) in stage I, liner carriers coordinate with each other to obtain a centralized opti ¼ f mization solution for all related liner carriers. Let x xkm ij g denote the optimal solution of model P. In stage II, we aim to determine the perceived values of empty container at different ports, based on the optimal solution of model P. Firstly, we determine the empty container exchange costs ({costj}) at the deficit ports, which are paid to liner carriers for delivering . The empty container exchange costs at the deficit ports can be empty containers and following the optimal solution x regarded as the parameters, which need to be properly determined in our model presented in stage II. The unit profits at the surplus ports can be calculated with respect to the empty container exchange costs at the deficit ports. Generally, a single liner carrier aims to maximize (or minimize) its own profit (cost). When the empty container exchange costs at the deficit ports are determined, we assume that the cost associated with liner carrier k mainly consists of three terms: (i) the cost for transporting empty containers from the surplus ports of liner carrier k, which is described as P P P km i2Sk m2L j2Wm ðkij xij Þ; (ii) the empty container exchange cost paid by liner carrier k to other liner carriers, which is forh i P P ; (iii) the empty container exchange cost paid to liner carrier k, which is given as mulated as j2Wk costj nkj i2Sk xkk ij P P P km i2Sk m–k;m2L j2Wm ðcost j xij Þ. Then, the optimal decision making problem for liner carrier k to maximize its profit, can be formulated as a linear programming model (denoted by Pk), shown as follows:
max
XX X h
ðcost j kij Þ xkm ij
i
ð6Þ
i2Sk m2L j2Wm
subject to
XX
k xkm ij 6 ni ;
8 i 2 Sk ;
ð7Þ
m2L j2Wm
X k xkk ij 6 nj ;
8 j 2 Wk ;
ð8Þ
i2Sk
xkm ij P 0;
8 i 2 Sk ; 8 j 2 Wm ; 8 m 2 L:
ð9Þ
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The objective function (6) represents the profit associated with liner carrier k plus
P
j2Wk
h
i costj ðnkj Þ ; which is a con-
stant when the empty container exchange costs at the deficit ports are determined. Constraints (7) and (8) ensure that empty containers of liner carrier k originated from (or delivered to) all the surplus (or deficit) ports cannot be larger than their surplus (or deficit). Clearly, not all empty containers of liner carrier k are used to satisfy the deficit ports of liner carrier k, since some empty containers of liner carrier k may be delivered to the deficit ports of other liner carriers, and some empty containers of other liner carriers may be delivered to the deficit ports of liner carrier k. From a single liner carrier point of view, there is no necessity for liner carrier k to deliver its own empty containers to the deficit ports of other liner carriers. As a result, liner carrier k may keep some residual empty containers at its surplus ports. In practice, this liner carrier may lease them off in order not to keep the additional empty containers and avoid the repositioning cost. For simplicity, this paper does not consider leasing empty containers. Actually, empty containers will be delivered if found profitable, as shown in the following complementary slackness conditions corresponding to the primal problem Pk and the dual problem of Pk. Hence, constraints (7) of Pk are different from constraints (3) of P. Constraints (9) denote xkm ij as a nonnegative variable. (the optimal solution of the model P) to the solution space of the model Pk. In the folk denote the projection of x Let x k is an optimal solution of lowing, we aim to identify the empty container exchange costs at the deficit ports, in order that x the model Pk. According to constraints (3) and (4), we can obtain that
XX
k xkm ij ¼ ni ;
8 i 2 Sk
ð10Þ
m2L j2Wm
X X X k xkk xmk ij þ ij ¼ nj ; i2Sk
8 j 2 Wk
ð11Þ
m–k2Li2Sm
k is a feasible solution of the model Pk. Clearly, x Next, we demonstrate the use of the inverse optimization technique (Ahuja and Orlin, 2001) to determine the empty container exchange costs at the deficit ports. The inverse optimization problems have been studied in some areas such as traffic equilibrium (Dial, 2000; Ahuja and Orlin, 2001) and network design for liner carrier alliances (Agarwal and Ergun, 2010; Zheng et al., 2015). n o n o For liner carrier k, let wk ¼ wki : wki P 0; i 2 Sk and gk ¼ gkj : gkj P 0; 8 j 2 Wk denote the dual variables associated with constraints (7) and (8), respectively. Then the dual problem of Pk, DPk can be described as,
min
X X ðnki wki Þ ðnkj gkj Þ i2Sk
ð12Þ
j2Wk
subject to
wki þ gkj P costj kij ; wki P costj kij ; wki P 0;
8 i 2 Sk ; 8 j 2 Wk ;
8 i 2 Sk ; 8 j 2 Wm ; 8 m–k; m 2 L;
8 i 2 Sk ;
gkj P 0; 8 j 2 Wk :
ð13Þ ð14Þ ð15Þ ð16Þ
km In the above DPk, constraints (13) and (14) are the dual feasibility conditions for variables fxkk ij g and fxij g, respectively. k ) and the dual solution One form of the linear programming optimality conditions states that the primal solution (x k is a feasible solution of Pk and (wk, gk) is a feasible solution of DPk, and together ((wk, gk)) are optimal for their problems if x they satisfy the complementary slackness conditions as follows: k k kk (1) If x ij > 0 (8i 2 Sk ; 8j 2 Wk ), then wi þ gj ¼ cost j kij . k km (2) If x ij > 0 (8i 2 Sk ; 8j 2 Wm ; 8m–k; m 2 L), then wi ¼ cost j kij . P kk k k (3) If i2Sk xij < nj (8j 2 Wk ), then gj ¼ 0. P P k k xkm (4) If m2L j2Wm ij < ni (8i 2 Sk ), then wi ¼ 0.
According to Eq. (10), the fourth complementary slackness condition can be omitted. The inverse optimization problem for liner carrier k aims to find a feasible dual solution and a cost vector satisfying the above complementary slackness conditions. Let Ik denote the inverse optimization problem for liner carrier k. The following theorem shows the non-uniqueness of the solution of Ik.
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g ¼ fcostj þ cg is also a feasible cost vector, where c is an arbitrary Theorem 1. If a cost vector cost ¼ fcostj g satisfies Ik, then cost nonnegative number. Proof. For the cost vector cost ¼ fcostj g, we assume that a dual solution ðwk ; gk Þ satisfies all constraints of Ik (i.e., all constraints of DPk and the complementary slackness conditions). Then, we can introduce another feasible dual solution ðwk þ c; gk Þ, where wk þ c ¼ fwki þ cg and c is an arbitrary nonnegative number. For this feasible dual solution, we can find g ¼ fcostj þ cg, since all constraints of Ik can be satisfied.h a feasible cost vector cost From Theorem 1, we can define the objective function of Ik as the minimization of the empty container exchange costs associated with liner carrier k, described as follows:
min
XX X
X XX
ðxkm ij cost j Þ þ
i2Sk m2L j2Wm
ðxmk ij cost j Þ
ð17Þ
m–k;m2Li2Sm j2Wk
Now, we wish to determine a common cost vector satisfying the inverse problems for all related liner carriers. Namely,
INVP :
[k I:
ð18Þ
k2L
n o Let Ak denote the set of fði; jÞj8i 2 Sk ; 8j 2 Wk g, and let Ak be the set of ði; jÞj xkk ij > 0; 8i 2 Sk ; 8j 2 Wk . Let Bkm denote the n o set of fði; jÞj8i 2 Sk ; 8j 2 Wm ; 8m–k; m 2 Lg, and let Bkm represent the set of ði; jÞj xkm ij > 0; 8i 2 Sk ; 8j 2 Wm ; 8m–k; m 2 L . Let P k k xkk Ck be the set of fj i2Sk ij < nj ; 8j 2 Wk g. Clearly, Ak # Ak , Bkm # Bkm and Ck # Wk . Similar to I , the optimization problem INVP can be formulated as follows:
min
XXX X
ðxkm ij cost j Þ
ð19Þ
k2L i2Sk m2L j2Wm
subject to
wki þ gkj cost j P kij ; wki costj P kij ;
8 ði; jÞ 2 Bkm n Bkm ; 8 k 2 L; 8 m–k; m 2 L;
wki þ gkj cost j ¼ kij ; wki costj ¼ kij ;
8 ði; jÞ 2 Ak n Ak ; 8 k 2 L;
8 ði; jÞ 2 Ak ; 8 k 2 L;
8 ði; jÞ 2 Bkm ; 8 k 2 L; 8 m–k; m 2 L;
ð20Þ ð21Þ ð22Þ ð23Þ
gkj ¼ 0; 8 j 2 Ck ; 8 k 2 L;
ð24Þ
8 i 2 Sk ; 8 k 2 L;
ð25Þ
wki P 0;
gkj P 0; 8 j 2 Wk ; 8 k 2 L; costj P 0;
8 j 2 W:
ð26Þ ð27Þ
The objective function (19) minimizes the total involved empty container exchange costs. Constraints (20) and (21) are equivalent to constraints (13) and (14) in the model DPk. Constraints (22–24) correspond to the first three complementary slackness conditions, respectively. As shown in the objective function (6) of Pk, ðcostj kij Þ can be regarded as the unit profit associated with port pair (i, j) for transporting one empty container from surplus port i to deficit port j. According to the first two complementary slackness conditions, we have costj kij P 0 if the empty container flow from surplus port i to deficit port j is positive. It seems that empty containers will be delivered if found profitable. Furthermore, we can obtain the following theorem. Theorem 2. For any particular surplus port i of liner carrier k, its associated unit profits for different deficit ports of other liner carriers are identical and equal to wki , as long as these deficit ports will be served by surplus port i for delivering empty containers. Proof. For any deficit port j ð8j 2 W n W k Þ served by surplus port i ð8i 2 Sk Þ, the empty container flow from surplus port i to this deficit port is larger than zero. According to the second complementary slackness condition, we have wki ¼ costj kij .h Actually, the unit profit for delivering empty containers from surplus port i of liner carrier k to the deficit ports of liner carrier k is also equal to wki , as shown in Section 4.2. Then, we can define wki (i.e., costj kij ) as the unit profit at surplus port i for delivering one empty container to any possible deficit port.
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4. Numerical experiments Our models are efficiently solved by using CPLEX implemented in a Windows 7 environment. Numerical experiments are performed on a 3.4 GHz Dual Core PC with 4 GB of RAM. 4.1. Data description We perform our computational experiments on an Asia–Europe–Oceania shipping service network consisting of 46 ports (Meng and Wang, 2011; Wang and Meng, 2011, 2012; Zheng et al., 2015), as shown in Fig. 1. In order to investigate the coordination among different numbers of liner carriers, this paper considers four liner carriers: liner 1, liner 2, liner 3 and liner 4. The OD demands for different liner carriers are derived from real data (with modification due to the commercial confidentiality). In order to obtain the average transportation cost kij for shipping one TEU empty container from port i to port j, we mainly consider the bunker cost of a 5000-TEU ship with a common sailing speed (e.g., 20 knots), since the bunker cost is a 3000 ¼ 0:03), major component of the ship operating cost. Following Brouer et al. (2014), we assume kij ¼ 0:03 Disij (i.e., 205000 where Disij is the distance between port i and port j. 4.2. Results analysis Firstly, we investigate the benefit of liner carrier coordination. For simplicity, we show the results for two liner carriers. In order to explore the benefit of liner carrier coordination, two cases, i.e., case I (two liner carriers do not coordinate) and case II (two liner carriers coordinate) are mainly investigated. In case I, empty containers are not exchanged between two liner carriers. In this case, constraints (3) and (4) of the model P can be rewritten as follows:
X
k xkk ij ¼ ni ;
8 i 2 Sk ; 8 k 2 L
ð28Þ
j2Wk
X k xkk ij ¼ nj ;
8 j 2 Wk ; 8 k 2 L
ð29Þ
i2Sk
As shown in Table 1, it is clear that, in case I (i.e., two liner carriers do not coordinate), the total transportation costs for two liner carriers are 1.64 million USD and 1.38 million USD, respectively. When two liner carriers coordinate (i.e., case II),
Fig. 1. Ports in an Asia–Europe–Oceania shipping service network.
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J. Zheng et al. / Transportation Research Part E 83 (2015) 158–169
the total transportation costs for two liner carriers become 1.18 million USD and 1.69 million USD, respectively. The overall transportation cost for two liner carriers can be reduced by 0.15 (i.e., (1.64 + 1.38) (1.18 + 1.69) = 0.15) million USD when they coordinate. Table 1 also shows the total empty container exchange cost paid to liner 1 and liner 2, respectively. We can find that liner 1 should pay 0.36 (i.e., 1.88 1.52 = 0.36) million USD to liner 2. Then, in case II, the total costs for liner 1 and liner 2 become 1.54 million USD and 1.33 million USD, respectively. Evidently, two liner carriers can benefit from the coordination with each other. Next, we show the results for the empty container exchange cost at each deficit port and the unit profit for delivering empty containers from each surplus port, as shown in Tables 2 and 3, respectively. In addition, the results for empty container allocation for the surplus ports of two liner carriers are shown in Tables A1 and A2, provided in Appendix A. The results for empty container allocation together with the empty container exchange costs can be used to verify the unit profits at any surplus port for delivering empty containers to different deficit ports. Table 2 shows the results for the empty container exchange costs at the deficit ports. As compared with other deficit ports, Bremerhaven has the lowest value of empty container exchange cost (i.e., 4.56 USD), and Tokyo has the largest value of empty container exchange cost (i.e., 339.87 USD). Moreover, the empty container exchange costs at the deficit ports in Europe are lower than 15 USD, since many empty containers are accumulated at the surplus ports in Europe, because of the imbalance between imports and exports for Asia-Europe trade. According to Theorem 2, we have proved that the unit profits at any particular surplus port of a liner carrier for serving different deficit ports of other liner carriers are identical. Here, we will further show that the unit profits for delivering empty containers from any particular surplus port of a single liner carrier to different deficit ports of this liner carrier are also identical. As shown in Table A2, Melbourne, Chittagong, Aqabah, Port Klang, Salalah and Jeddah are the surplus ports of liner 2, all of which only serve the deficit ports of liner 2. For instance, Brisbane and Sydney are served by Melbourne, and the transportation distances from Melbourne to Brisbane and Sydney are 1097 nautical miles and 582 nautical miles, respectively. According to Table 2, the unit profit for delivering empty containers from Melbourne to Brisbane is 260.91 USD (i.e., 293.82 0.03 1097 = 260.91), and the unit profit for delivering empty containers from Melbourne to Sydney is also P k 260.91 USD (i.e., 278.37 0.03 582 = 260.91). It seems that we also have gkj ¼ 0 when i2Sk xkk ij ¼ nj (8j 2 Wk ) is satisfied, P kk k xij ¼ nj (8j 2 Wk ) is satisfied, the empty consimilar to the third complementary slackness condition. Actually, when i2Sk tainer exchange cost at deficit port j of liner carrier k, which is dependent on gkj , is paid to liner carrier k herself. In other words, various values of gkj do not impact the coordination and interaction among different liner carriers. In order to minimize the objective function of INVP (19), gkj ¼ 0 can be obtained. Furthermore, when any particular surplus port of a single liner carrier k serves different deficit ports of different liner carriers including liner carrier k, one can find that the unit profits at this surplus port for delivering empty containers to different deficit ports are also identical. Actually, we obtain a
Table 1 Comparison between two cases. Total transportation cost (million USD)
Case I Case II
Total empty container exchange cost (million USD)
Liner 1
Liner 2
To liner 1
To liner 2
1.64 1.18
1.38 1.69
– 1.52
– 1.88
Table 2 Empty container exchange costs at the deficit ports. Deficit ports
Empty container exchange costs (USD)
Deficit ports
Empty container exchange costs (USD)
Brisbane Fremantle Adelaide Sydney Zeebrugge Le Havre Bremerhaven Hong Kong Cochin Chennai Nhava Sheve Jakarta Tokyo Nagoya Yokohama Kobe Pusan Kwangyang
293.82 205.2 245.49 278.37 9.57 14.7 4.56 308.52 210.78 216.42 204.78 258.09 339.87 333.09 339.51 328.92 319.41 320.67
Port Klang Dalian Xingang Qingdao Ningbo Shanghai Karachi Manila Singapore Yantian Xiamen Chiwan Colombo Kaohsiung Leam Chabang Jebel Ali Thamesport Ho Chi Minh
258.42 317.64 323.64 322.2 326.13 334.17 207.78 304.95 264.72 309.72 317.13 309.72 218.34 317.73 278.22 186.45 12.03 284.1
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Table 3 The unit profits for the surplus ports. Surplus ports
The unit profits (USD)
Surplus ports
The unit profits (USD)
Adelaide Antwerp Aqabah Brisbane Chennai Chittagong Colombo Dalian Fremantle Hamburg Hong Kong Jakarta Jebel Ali Jeddah Karachi Kobe
245.49 7.44 118.32 293.82 216.42 219.09 218.34 317.64 205.2 0 308.52 258.09 186.45 135.54 207.78 328.92
Kwangyang Le Havre Manila Melbourne Nagoya Nhava Sheve Port Klang Pusan Qingdao Rotterdam Salalah Singapore Sokhna Southampton Sydney Zeebrugge
320.67 14.7 304.95 260.91 333.09 204.78 258.42 319.41 322.2 7.59 171.78 264.72 97.32 15.15 278.37 9.57
unique profit for any surplus port when delivering one empty container to any possible deficit port. In Table 3, we show the unit profit at each surplus port for providing empty containers. Clearly, Hamburg has the lowest value of the unit profit, which is equal to 0 USD, and Nagoya has the largest value of the unit profit (i.e., 333.09 USD). Compared with other surplus ports, Hamburg is the farthest surplus port for delivering empty containers to the deficit ports, as shown in Fig. 1. Compared with other deficit ports, the deficit ports in Japan (e.g., Tokyo, Yokohama, etc.) have a large value of the empty container exchange cost, as shown in Table 2. Hence, as the nearest surplus port to Yokohama, Nagoya can obtain the largest value of the unit profit for delivering empty containers to Yokohama. In practice, when the empty container supply does not meet the demand, some liner carriers may lease empty containers. As compared with the container leasing prices provided by the container leasing companies, the perceived values of empty container (i.e., the unit profits and the empty container exchange costs) at different ports may have an impact on pricing in the container leasing market. For any particular surplus port, liner carriers would like to reposition empty containers if the unit profit at this port is larger than the off-leasing price. Otherwise, liner carriers would lease off their empty containers in order to avoid the repositioning costs. For any particular deficit port, liner carriers would like to lease in empty containers if the empty container exchange cost at this deficit port is larger than the leasing-in price. Otherwise, the coordination among liner carriers for exchanging empty containers would be favorable. Now, we investigate the stability of liner carrier coordination on empty container allocation by considering the coordination among different numbers of liner carriers. Table 4 shows the results of all combinations for the coordination among four liner carriers. The second column in Table 4 shows the different coordination cases. For example, (1, 2, 3) means that liner 1, liner 2 and liner 3 coordinate with each other. In order for easy comparison, Table 4 also shows the individual transportation cost in the non-coordination case. Evidently, one can find that any two of four carriers can benefit from the coordination with each other, as compared with the non-coordination case. In other words, the coordination formation is stable when the number of liner carriers is 2. When the number of liner carriers is 3 or 4, the coordination formation among liner carriers is also quite stable. This is because there is no motivation for some liner carriers to collude, in order to decrease their costs. In addition, results for cost allocation by using the Shapley value are also shown in Table 4. Clearly, our results are slightly different with the Shapley value. Table 4 Results for the coordination among different numbers of liner carriers. Total cost (million USD) Liner 1
Liner 2
Liner 3
Liner 4
Two carriers
(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)
1.54 1.63 1.63 – – –
1.33 – – 1.34 1.31 –
– 1.29 – 1.19 – 1.29
– – 1.92 – 1.82 1.92
Three carriers
(1, 2, 3) (1, 2, 4) (1, 3, 4) (2, 3, 4)
1.55 1.58 1.37 –
1.31 1.27 – 1.30
1.21 – 1.57 1.22
– 1.85 1.80 1.83
Four carriers Non-coordination Shapley value
(1, 2, 3, 4)
1.62 1.64 1.5
1.16 1.38 1.15
1.27 1.80 1.33
1.90 2.50 1.97
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5. Conclusions This paper has investigated an empty container allocation problem considering the coordination among liner carriers for exchanging empty containers. We further measure the perceived values of empty container at different ports. To solve our problems, we propose a two-stage optimization method. Numerical implementations show the following conclusions: (i) liner carriers can benefit from the coordination with each other; (ii) empty containers are delivered if found profitable; (iii) the coordination formation among liner carriers is found to be stable; (iv) the perceived values of empty container at different ports are measured. Furthermore, by comparing the perceived values of empty container at different ports with the container leasing prices provided by the container leasing companies, we hope our work is helpful for pricing in the container leasing market. As shown in our assumptions, there are some limitations in our work: (i) travel times for delivering empty containers are neglected; (ii) the weekly container demands are fixed and given in advance; (iii) we only consider the average transportation cost when repositioning empty containers; (iv) each liner carrier is assumed to have sufficient space for repositioning empty containers. There are some research issues we will investigate in the future. Firstly, we will extend our work by considering the weekly dependent container demands, since the container demands vary from week to week. Secondly, we will study the repositioning of empty containers and the routing of laden containers in a liner shipping network, by considering ship capacity constraints, ship operating cost, bunker cost, inventory cost and container handling cost, etc. Thirdly, we will investigate the impact of the container leasing prices on our results, in order to better understand the container leasing market. Acknowledgements We would like to thank anonymous referees for their useful comments, which significantly improve the presentation of the paper. This research is supported by the National Basic Research Program of China (2012CB725400), the National Natural Table A1 Empty container allocation for the surplus ports of liner 1. Surplus ports
Surplus containers
Brisbane
87
Adelaide
192
Melbourne
202
Sydney
191
Chittagong
112
Zeebrugge
107
Sokhna Hamburg Nhava Sheve
1433 527 96
Jakarta
296
Aqabah
521
Rotterdam
280
Salalah
381
Karachi
121
Manila
66
Jeddah
726
Singapore
146
Colombo
126
Jebel Ali
385
Southampton
1685
Liner carriers Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
Deficit ports with delivered empty containers Brisbane (87) Adelaide (192) Brisbane (202) Sydney (191) Ho Chi Minh (112) Zeebrugge (85), Manila (22) Ningbo (386), Shanghai (857) Yantian (190) Bremerhaven (24), Ho Chi Minh (52) Bremerhaven (148), Thamesport (303) Nhava Sheve (96) Jakarta (155), Manila (141) Yokohama (521) Hong Kong (59) Tokyo (221) Cochin (4), Nhava Sheve (229), Karachi (148) Karachi (121) Manila (66) Cochin (726) Xiamen (146) Colombo (126) Karachi (204), Jebel Ali (181) Xiamen (295) Ningbo (1390)
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Table A2 Empty container allocation for the surplus ports of liner 2. Surplus ports
Surplus containers
Liner carriers
Deficit ports with delivered empty containers
Fremantle
1296
Fremantle (19) Adelaide (475), Sydney (436), Jakarta (366)
Melbourne
821
Chittagong
51
Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner
Antwerp Sokhna
923 1796
Le Havre
800
Hamburg
880
Hong Kong
56
Chennai
551
Nagoya
114
Kobe
246
Aqabah
284
Pusan
290
Kwangyang
856
Port Klang
174
Rotterdam
594
Dalian
1531
Qingdao
1431
Salalah
361
Jeddah
275
Southampton
1435
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
Brisbane (547), Sydney (274) Manila (25), Ho Chi Minh (26) Cochin (18), Kaohsiung (224), Leam Chabang (160), Ho Chi Minh (134) Manila (59), Yantian (76), Colombo (132), Kaohsiung (120) Port Klang (41), Ningbo (431), Chiwan (398) Singapore (543), Leam Chabang (383) Le Havre (5), Chiwan (626) Xiamen (169) Thamesport (4) Bremerhaven (876) Chiwan (56) Chennai (22), Yokohama (20), Port Klang (221) Chiwan (288) Nagoya (48), Yokohama (46) Yokohama (20) Yokohama (41), Kobe (162) Tokyo (43) Yokohama (284) Pusan (224) Shanghai (66) Kwangyang (249) Shanghai (607) Singapore (174) Hong Kong (25), Xiamen (569) Dalian (59), Xingang (95), Shanghai (812) Xingang (565) Qingdao (201), Shanghai (90) Shanghai (1140) Cochin (361) Nhava Sheve (275) Hong Kong (419), Tokyo (5), Yantian (583) Karachi (428)
Science Foundation of China (71501021, 61304179), the Fundamental Research Funds for the Central Universities (3132015069), and Research Fund for the Doctoral Program of Higher Education of China (20130009120001). Appendix A The results for empty container allocation for the surplus ports of two liner carriers are shown in Tables A1 and A2, respectively. References Agarwal, R., Ergun, O., 2010. Network design and allocation mechanisms for carrier alliances in liner shipping. Oper. Res. 58 (6), 1726–1742. Ahuja, R.K., Orlin, J.B., 2001. Inverse optimization. Oper. Res. 49 (5), 771–783. Bell, M.G.H., Liu, X., Angeloudis, P., Fonzone, A., Hosseinloo, S.H., 2011. A frequency-based maritime container assignment model. Transp. Res. Part B 45 (8), 1152–1161. Bell, M.G.H., Liu, X., Rioult, J., Angeloudis, P., 2013. A cost-based maritime container assignment model. Transp. Res. Part B 58, 58–70. Bonney, J., 2004. Changing the landscape. J. Commer. (April 26) Braekers, K., Janssens, G.K., Caris, A., 2011. Challenges in managing empty container movements at multiple planning levels. Transp. Rev. 31 (6), 681–708. Brouer, B.D., Alvarez, J.F., Plum, C.E.M., Pisinger, D., Sigurd, M.M., 2014. A base integer programming model and benchmark suite for liner-shipping network design. Transp. Sci. 48 (2), 281–312. Brouer, B.D., Pisinger, D., Spoorendonk, S., 2011. Liner shipping cargo allocation with repositioning of empty containers. INFOR 49 (2), 109–124. Cheung, R.K., Chen, C.Y., 1998. A two-stage stochastic network model and solution methods for the dynamic empty container allocation problem. Transp. Sci. 32 (2), 142–162. Christiansen, M., Fagerholt, K., Nygreen, B., Ronen, D. 2007. Maritime Transportation. Barnhart, C., Laporte, G. (Eds.), Handbook in Operations Research & Management Science, vol. 14.
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Contents lists available at ScienceDirect
Transportation Research Part E journal homepage: www.elsevier.com/locate/tre
Empty container exchange among liner carriers Jianfeng Zheng a,⇑, Zhuo Sun a, Ziyou Gao b a b
Transportation Management College, Dalian Maritime University, Dalian 116026, China School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China
a r t i c l e
i n f o
Article history: Received 22 June 2015 Received in revised form 21 September 2015 Accepted 22 September 2015
Keywords: Empty container allocation Value of empty container Two-stage optimization method Inverse optimization
a b s t r a c t In an attempt to reduce the empty container repositioning costs, this paper studies an empty container allocation problem considering the coordination among liner carriers. We further measure the perceived values of empty container at different ports. The perceived values of empty container at the surplus (deficit) ports are described by the profits (empty container exchange costs paid) for delivering empty containers. To solve our problems, we propose a two-stage optimization method. In stage I, liner carriers are guided to pursue a centralized optimization solution of empty container allocation for all related liner carriers. In stage II, the inverse optimization technique is used to determine the empty container exchange costs, which are paid to liner carriers for exchanging empty containers and following the centralized optimization solution. The profits at the surplus ports are calculated with respect to the empty container exchange costs at the deficit ports. Finally, numerical experiments on an Asia–Europe–Oceania shipping service network are discussed. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction In the liner shipping industry, empty container repositioning is a challenge for liner carriers due to the high costs. Since 1993, empty container movements have constituted about 20% of the total ocean container movements (Song and Dong, 2011). In 2003, the repositioning cost was up to $11 billion (Bonney, 2004), and that in 2010 was about $23.4 billion (Drewry, 2011; Tran and Haasis, 2015). Song et al. (2005) estimated that the repositioning cost accounts for 27% of the total world fleet running cost. Because of the trade imbalances between the major trading regions, empty container movements cannot be avoided completely. However, minimizing these costly activities would considerably reduce the operating costs of liner carriers. In an attempt to reduce the cost on repositioning empty containers, this paper proposes an empty container allocation problem considering the coordination among liner carriers, where empty containers of any single liner carrier can be repositioned to serve the needs of other liner carriers. Hence, empty containers are exchanged among liner carriers. Similar to slot exchange agreements between liner carriers in a strategic alliance, empty container exchange agreements can be signed between liner carriers to reduce the repositioning costs of liner carriers. Generally, empty container movements do not generate revenue for liner carriers. In order to motivate liner carriers to follow a centralized optimization solution of empty container allocation for all related liner carriers, the extra incentives should be introduced to guide liner carriers. Generally, these incentives can be described by the profits (costs paid) for delivering empty containers from (to)
⇑ Corresponding author. E-mail address: [email protected] (J. Zheng). http://dx.doi.org/10.1016/j.tre.2015.09.007 1366-5545/Ó 2015 Elsevier Ltd. All rights reserved.
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the surplus (deficit) ports, which have a surplus (deficit) of empty containers. This motivates us to measure the perceived values of empty container at different ports. There have been many studies related to the empty container allocation problem or the empty container repositioning issue. Crainic et al. (1993) developed two dynamic deterministic formulations and a stochastic formulation for empty container allocation in a land distribution and transportation system. Cheung and Chen (1998) investigated the dynamic empty container allocation problem, which is formulated as a two-stage stochastic network model. A stochastic quasigradient method and a stochastic hybrid approximation procedure were applied to solve the problem. Erera et al. (2009) developed a robust optimization framework for dynamic empty repositioning problems modeled using time–space networks. Li et al. (2004) discussed empty container management at a port and derived the optimal pairs of critical policies, (U, D) for this port. Namely, if the number of empty containers at this port is less than U, empty containers are imported up to U, or empty containers are exported down to D if the number of empty containers at this port is larger than D. Later, Li et al. (2007) extended this problem by considering multiple ports. Jula et al. (2006) studied empty container movements by optimizing the empty container reuse. The dynamic empty container reuse was modeled and optimization techniques were developed to optimize the empty container operations. Lam et al. (2007) presented an approximate dynamic programming approach, in order to obtain the effective empty container relocation strategies. Feng and Chang (2008) investigated the repositioning of empty containers for intra-Asia liner shipping. Song and Carter (2009) studied general empty container balancing strategies depending on whether shipping lines are coordinating the container flows over different routes and whether they are willing to share container fleets. Moon et al. (2010) proposed an empty container repositioning problem considering leasing and purchasing. To address this problem, they presented a mixed-integer linear optimization model and developed a genetic algorithm to solve it. Shintani et al. (2010) analyzed the possibility to save the container fleet management costs in repositioning empty containers by using foldable containers. Later, Moon et al. (2013) compared the foldable containers with the standard containers on the cost for repositioning empty containers. Numerical experiments demonstrated the economic feasibility of foldable containers. Di Francesco et al. (2009) addressed an empty container repositioning problem under uncertainty, where the historical data were inappropriate for estimating uncertain parameters. In order to solve this problem, a time-extended multi-scenario optimization model was developed. Later, Di Francesco et al. (2013) studied an empty container repositioning problem under uncertain port disruptions. Long et al. (2012, 2015) investigated an empty container repositioning problem with uncertainties, by using a sample average approximation method. Song and Dong (2011) discussed an empty container repositioning policy with flexible destination ports. Bell et al. (2011, 2013) proposed two types of container assignment models (i.e., a frequency-based container assignment model and a cost-based container assignment model), in which both laden containers and empty containers were considered. Recently, Wang et al. (2015) extended these two container assignment models by proposing several profit-based container assignment models. In addition, readers can refer to the references in two reviews on empty container repositioning (Braekers et al., 2011; Song and Dong, 2015). Furthermore, some researchers explored the combined optimization problems in liner shipping with empty container repositioning. Shintani et al. (2007), Meng and Wang (2011), Song and Dong (2013) investigated the liner shipping network design problem, as well as considering the repositioning of empty containers. Dong and Song (2009) addressed the joint problem of container fleet sizing and empty container repositioning. Brouer et al. (2011) studied the cargo allocation problem with the repositioning of empty containers. Song and Dong (2012) investigated cargo routing, together with empty container repositioning. For more optimization problems related to ship routing and scheduling in liner shipping, please refer to some review papers (Christiansen et al., 2004, 2013; Meng et al., 2014). To the best of our knowledge, there is currently no optimization model on the empty container allocation problem considering the coordination among liner carriers. Le (2003) studied a related problem based on a neutral Internet-based information exchange platform, which may facilitate empty container reuse and sharing empty containers among liner carriers. Theofanis and Boile (2009) mentioned that empty containers of a liner carrier can be used to match the needs of other liner carriers. Both of these works discussed the empty container repositioning strategies from a qualitative point of view. Obviously, the coordination among liner carriers increases the flexibility of an empty container repositioning system by exchanging empty containers among liner carriers and offers an opportunity to reduce the repositioning costs of liner carriers. However, the coordination formation among liner carriers is challenging. Generally, the goal of a liner carrier is the maximization of its own profit (or minimization of its own cost). Furthermore, some liner carriers may collude to obtain a larger profit (or a lower cost), as compared with the coordination among more liner carriers. Hence, the coordination stability among liner carriers is also a challenge. This paper aims to resolve these problems in the study of empty container allocation considering the coordination among liner carriers. Furthermore, we will measure the perceived values of empty container at different ports, under the coordination among liner carriers. The rest of this paper is organized as follows. Notation, assumptions and problem description are described in Section 2. A two-stage optimization method is presented in Section 3. Numerical results are given in Section 4. The conclusions are shown in Section 5. 2. Notation, assumptions and problem description Some mathematical notations have to be defined in order to facilitate description and formulation of the problem.
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Parameters L Set of liner carriers P Set of ports S Set of surplus ports Sk Set of surplus ports for liner carrier k 2 L W Set of deficit ports Wk Set of deficit ports for liner carrier k 2 L Qk Set of OD (Origin-Destination) demands of liner carrier k 2 L qkij Weekly number of containers of liner carrier k transported from origin port i to destination port j, i.e., the OD demand kij The average transportation cost for delivering one empty container from port i to port j Decision variables xkm Weekly number of empty containers transported from surplus port i of liner carrier k 2 L to deficit port j of liner ij carrier m 2 L costj The empty container exchange cost for providing one TEU (Twenty-foot Equivalent Unit) empty container at deficit port j 2 W
Generally, shippers will choose the proper liner carriers for delivering their containers, and they will sign a short-term or long-term contract with their chosen liner carriers (Zheng et al., 2015). As described in Christiansen et al. (2007), it is not unusual that some liner carriers do between 80% and 95% of their business under the long-term contracts, most of which are negotiated once a year between liner carriers and shippers. According to the fixed container demands of different liner carriers, the number of empty containers accumulated at the surplus ports or lacked at the deficit ports for different liner carriers can be obtained. Generally, a practical ship route (i.e., port calling sequence by a ship) often maintains a weekly service frequency, and the weekly container demands of different liner carriers can be used to determine the distribution of empty containers at different ports. In order to simplify our problem, we assume that the weekly container demands of different liner carriers are fixed and given in advance. Given the weekly OD demand from origin port i to destination port j of liner carrier k, qkij ð8qkij 2 Qk Þ, let nki denote the difference between the incoming and outgoing container flow for liner carrier k at port i, calculated as follows:
nki ¼
X j2P
qkji
X qkij ;
8 k 2 L; 8i 2 P
ð1Þ
j2P
Clearly, nki represents the empty container supply or demand of liner carrier k at port i. When nki takes a positive value, port i is a surplus port of liner carrier k, and port i is a deficit port of liner carrier k, when its value is negative. Generally, the empty container allocating processes are typically time consuming and depend on the space available on vessels, both of which are not considered in this paper. Namely, we assume: (i) travel times for delivering empty containers are neglected; (ii) each liner carrier has sufficient space for repositioning empty containers. For simplicity, this paper mainly considers the average transportation cost for delivering empty containers from the surplus ports to the deficit ports. Note that the empty container repositioning is constrained by vessel spare capacities since laden containers have priority to be shipped (Dong and Song, 2009). In practice, empty containers may be transported by a less cost-efficient liner carrier due to capacity constraints. In other words, the transportation cost for transporting empty containers from some certain surplus ports to some certain deficit port may be various for different liner carriers, especially when considering the coordination among liner carriers. For instance, one liner carrier can buy some vessel slots from other liner carriers, leading to the different costs on transporting (laden and empty) containers between some certain ports for different liner carriers. In this paper, we assume that the average transportation cost for transporting one empty container from any surplus port to any deficit port is identical for different liner carriers. As mentioned before, one aim of this paper is to measure the perceived values of empty container at different ports, which are defined as follows. For a particular surplus port, its perceived value of empty container is described by the unit profit for delivering one empty container from this surplus port to any possible deficit port. For a particular deficit port, its perceived value of empty container is represented by the cost paid to liner carriers for delivering one empty container to this deficit port. This cost is termed ‘‘the empty container exchange cost” in this paper. Since empty container movements do not generate revenue for liner carriers, we assume that the unit profits for delivering empty containers from the surplus ports only depend on the average transportation cost for transporting empty containers and the empty container exchanges costs at the deficit ports, independent of any other aspects (e.g., container leasing prices). Because of the coordination among liner carriers, we assume that empty containers are repositioned in such a way that the total transportation cost on delivering empty containers for all related liner carriers is minimized, leading to a centralized optimization solution of empty container allocation. Then, we should determine the cost allocation (or profit sharing) among liner carriers. For a small number of participating liner carriers, using Shapley value (Shapley, 1953) would be a meaningful approach. Furthermore, this paper will measure the perceived values of empty container at different ports, based on the centralized optimization solution. To solve our problems, we propose a two-stage optimization method, as shown in Section 3.
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3. Two-stage optimization method Here, we propose a two-stage optimization method for solving our problems. In stage I, a centralized optimization solution of empty container allocation for all related liner carriers can be obtained, as shown in Section 3.1. In stage II, we aim to determine the perceived values of empty container at different ports based on the solution obtained in stage I, as shown in Section 3.2. 3.1. Stage I In stage I, we aim to determine the weekly number of empty containers delivered from surplus port i of liner carrier k 2 L to deficit port j of liner carrier m 2 L (i.e., xkm ij ) to obtain a centralized optimization solution of empty container allocation for all related liner carriers. The mathematical programming model (denoted by P) to pursue this centralized optimization solution can be formulated as follows:
XXX X
min
ðkij xkm ij Þ
ð2Þ
8 i 2 Sk ; 8 k 2 L;
ð3Þ
k2L m2L i2Sk j2Wm
subject to
XX
k xkm ij ¼ ni ;
m2L j2Wm
XX m xkm ij ¼ nj ;
8 j 2 Wm ; 8 m 2 L;
ð4Þ
k2L i2Sk
xkm ij P 0;
8 i 2 Sk ; 8 j 2 Wm ; 8 k; m 2 L:
ð5Þ
The objective function (2) minimizes the total transportation cost for delivering empty containers of all related liner carriers from the surplus ports to the deficit ports. Constraints (3) and (4) ensure that empty containers should be balanced after as a nonnegative variable. Note that the above formulation also allocating empty containers. Constraints (5) denote xkm ij accounts for the case k = m, where any single liner carrier delivers its own empty containers from its surplus ports to its deficit ports without considering the coordination among liner carriers. 3.2. Stage II Based on the optimization model P ((2–5)) in stage I, liner carriers coordinate with each other to obtain a centralized opti ¼ f mization solution for all related liner carriers. Let x xkm ij g denote the optimal solution of model P. In stage II, we aim to determine the perceived values of empty container at different ports, based on the optimal solution of model P. Firstly, we determine the empty container exchange costs ({costj}) at the deficit ports, which are paid to liner carriers for delivering . The empty container exchange costs at the deficit ports can be empty containers and following the optimal solution x regarded as the parameters, which need to be properly determined in our model presented in stage II. The unit profits at the surplus ports can be calculated with respect to the empty container exchange costs at the deficit ports. Generally, a single liner carrier aims to maximize (or minimize) its own profit (cost). When the empty container exchange costs at the deficit ports are determined, we assume that the cost associated with liner carrier k mainly consists of three terms: (i) the cost for transporting empty containers from the surplus ports of liner carrier k, which is described as P P P km i2Sk m2L j2Wm ðkij xij Þ; (ii) the empty container exchange cost paid by liner carrier k to other liner carriers, which is forh i P P ; (iii) the empty container exchange cost paid to liner carrier k, which is given as mulated as j2Wk costj nkj i2Sk xkk ij P P P km i2Sk m–k;m2L j2Wm ðcost j xij Þ. Then, the optimal decision making problem for liner carrier k to maximize its profit, can be formulated as a linear programming model (denoted by Pk), shown as follows:
max
XX X h
ðcost j kij Þ xkm ij
i
ð6Þ
i2Sk m2L j2Wm
subject to
XX
k xkm ij 6 ni ;
8 i 2 Sk ;
ð7Þ
m2L j2Wm
X k xkk ij 6 nj ;
8 j 2 Wk ;
ð8Þ
i2Sk
xkm ij P 0;
8 i 2 Sk ; 8 j 2 Wm ; 8 m 2 L:
ð9Þ
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The objective function (6) represents the profit associated with liner carrier k plus
P
j2Wk
h
i costj ðnkj Þ ; which is a con-
stant when the empty container exchange costs at the deficit ports are determined. Constraints (7) and (8) ensure that empty containers of liner carrier k originated from (or delivered to) all the surplus (or deficit) ports cannot be larger than their surplus (or deficit). Clearly, not all empty containers of liner carrier k are used to satisfy the deficit ports of liner carrier k, since some empty containers of liner carrier k may be delivered to the deficit ports of other liner carriers, and some empty containers of other liner carriers may be delivered to the deficit ports of liner carrier k. From a single liner carrier point of view, there is no necessity for liner carrier k to deliver its own empty containers to the deficit ports of other liner carriers. As a result, liner carrier k may keep some residual empty containers at its surplus ports. In practice, this liner carrier may lease them off in order not to keep the additional empty containers and avoid the repositioning cost. For simplicity, this paper does not consider leasing empty containers. Actually, empty containers will be delivered if found profitable, as shown in the following complementary slackness conditions corresponding to the primal problem Pk and the dual problem of Pk. Hence, constraints (7) of Pk are different from constraints (3) of P. Constraints (9) denote xkm ij as a nonnegative variable. (the optimal solution of the model P) to the solution space of the model Pk. In the folk denote the projection of x Let x k is an optimal solution of lowing, we aim to identify the empty container exchange costs at the deficit ports, in order that x the model Pk. According to constraints (3) and (4), we can obtain that
XX
k xkm ij ¼ ni ;
8 i 2 Sk
ð10Þ
m2L j2Wm
X X X k xkk xmk ij þ ij ¼ nj ; i2Sk
8 j 2 Wk
ð11Þ
m–k2Li2Sm
k is a feasible solution of the model Pk. Clearly, x Next, we demonstrate the use of the inverse optimization technique (Ahuja and Orlin, 2001) to determine the empty container exchange costs at the deficit ports. The inverse optimization problems have been studied in some areas such as traffic equilibrium (Dial, 2000; Ahuja and Orlin, 2001) and network design for liner carrier alliances (Agarwal and Ergun, 2010; Zheng et al., 2015). n o n o For liner carrier k, let wk ¼ wki : wki P 0; i 2 Sk and gk ¼ gkj : gkj P 0; 8 j 2 Wk denote the dual variables associated with constraints (7) and (8), respectively. Then the dual problem of Pk, DPk can be described as,
min
X X ðnki wki Þ ðnkj gkj Þ i2Sk
ð12Þ
j2Wk
subject to
wki þ gkj P costj kij ; wki P costj kij ; wki P 0;
8 i 2 Sk ; 8 j 2 Wk ;
8 i 2 Sk ; 8 j 2 Wm ; 8 m–k; m 2 L;
8 i 2 Sk ;
gkj P 0; 8 j 2 Wk :
ð13Þ ð14Þ ð15Þ ð16Þ
km In the above DPk, constraints (13) and (14) are the dual feasibility conditions for variables fxkk ij g and fxij g, respectively. k ) and the dual solution One form of the linear programming optimality conditions states that the primal solution (x k is a feasible solution of Pk and (wk, gk) is a feasible solution of DPk, and together ((wk, gk)) are optimal for their problems if x they satisfy the complementary slackness conditions as follows: k k kk (1) If x ij > 0 (8i 2 Sk ; 8j 2 Wk ), then wi þ gj ¼ cost j kij . k km (2) If x ij > 0 (8i 2 Sk ; 8j 2 Wm ; 8m–k; m 2 L), then wi ¼ cost j kij . P kk k k (3) If i2Sk xij < nj (8j 2 Wk ), then gj ¼ 0. P P k k xkm (4) If m2L j2Wm ij < ni (8i 2 Sk ), then wi ¼ 0.
According to Eq. (10), the fourth complementary slackness condition can be omitted. The inverse optimization problem for liner carrier k aims to find a feasible dual solution and a cost vector satisfying the above complementary slackness conditions. Let Ik denote the inverse optimization problem for liner carrier k. The following theorem shows the non-uniqueness of the solution of Ik.
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g ¼ fcostj þ cg is also a feasible cost vector, where c is an arbitrary Theorem 1. If a cost vector cost ¼ fcostj g satisfies Ik, then cost nonnegative number. Proof. For the cost vector cost ¼ fcostj g, we assume that a dual solution ðwk ; gk Þ satisfies all constraints of Ik (i.e., all constraints of DPk and the complementary slackness conditions). Then, we can introduce another feasible dual solution ðwk þ c; gk Þ, where wk þ c ¼ fwki þ cg and c is an arbitrary nonnegative number. For this feasible dual solution, we can find g ¼ fcostj þ cg, since all constraints of Ik can be satisfied.h a feasible cost vector cost From Theorem 1, we can define the objective function of Ik as the minimization of the empty container exchange costs associated with liner carrier k, described as follows:
min
XX X
X XX
ðxkm ij cost j Þ þ
i2Sk m2L j2Wm
ðxmk ij cost j Þ
ð17Þ
m–k;m2Li2Sm j2Wk
Now, we wish to determine a common cost vector satisfying the inverse problems for all related liner carriers. Namely,
INVP :
[k I:
ð18Þ
k2L
n o Let Ak denote the set of fði; jÞj8i 2 Sk ; 8j 2 Wk g, and let Ak be the set of ði; jÞj xkk ij > 0; 8i 2 Sk ; 8j 2 Wk . Let Bkm denote the n o set of fði; jÞj8i 2 Sk ; 8j 2 Wm ; 8m–k; m 2 Lg, and let Bkm represent the set of ði; jÞj xkm ij > 0; 8i 2 Sk ; 8j 2 Wm ; 8m–k; m 2 L . Let P k k xkk Ck be the set of fj i2Sk ij < nj ; 8j 2 Wk g. Clearly, Ak # Ak , Bkm # Bkm and Ck # Wk . Similar to I , the optimization problem INVP can be formulated as follows:
min
XXX X
ðxkm ij cost j Þ
ð19Þ
k2L i2Sk m2L j2Wm
subject to
wki þ gkj cost j P kij ; wki costj P kij ;
8 ði; jÞ 2 Bkm n Bkm ; 8 k 2 L; 8 m–k; m 2 L;
wki þ gkj cost j ¼ kij ; wki costj ¼ kij ;
8 ði; jÞ 2 Ak n Ak ; 8 k 2 L;
8 ði; jÞ 2 Ak ; 8 k 2 L;
8 ði; jÞ 2 Bkm ; 8 k 2 L; 8 m–k; m 2 L;
ð20Þ ð21Þ ð22Þ ð23Þ
gkj ¼ 0; 8 j 2 Ck ; 8 k 2 L;
ð24Þ
8 i 2 Sk ; 8 k 2 L;
ð25Þ
wki P 0;
gkj P 0; 8 j 2 Wk ; 8 k 2 L; costj P 0;
8 j 2 W:
ð26Þ ð27Þ
The objective function (19) minimizes the total involved empty container exchange costs. Constraints (20) and (21) are equivalent to constraints (13) and (14) in the model DPk. Constraints (22–24) correspond to the first three complementary slackness conditions, respectively. As shown in the objective function (6) of Pk, ðcostj kij Þ can be regarded as the unit profit associated with port pair (i, j) for transporting one empty container from surplus port i to deficit port j. According to the first two complementary slackness conditions, we have costj kij P 0 if the empty container flow from surplus port i to deficit port j is positive. It seems that empty containers will be delivered if found profitable. Furthermore, we can obtain the following theorem. Theorem 2. For any particular surplus port i of liner carrier k, its associated unit profits for different deficit ports of other liner carriers are identical and equal to wki , as long as these deficit ports will be served by surplus port i for delivering empty containers. Proof. For any deficit port j ð8j 2 W n W k Þ served by surplus port i ð8i 2 Sk Þ, the empty container flow from surplus port i to this deficit port is larger than zero. According to the second complementary slackness condition, we have wki ¼ costj kij .h Actually, the unit profit for delivering empty containers from surplus port i of liner carrier k to the deficit ports of liner carrier k is also equal to wki , as shown in Section 4.2. Then, we can define wki (i.e., costj kij ) as the unit profit at surplus port i for delivering one empty container to any possible deficit port.
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4. Numerical experiments Our models are efficiently solved by using CPLEX implemented in a Windows 7 environment. Numerical experiments are performed on a 3.4 GHz Dual Core PC with 4 GB of RAM. 4.1. Data description We perform our computational experiments on an Asia–Europe–Oceania shipping service network consisting of 46 ports (Meng and Wang, 2011; Wang and Meng, 2011, 2012; Zheng et al., 2015), as shown in Fig. 1. In order to investigate the coordination among different numbers of liner carriers, this paper considers four liner carriers: liner 1, liner 2, liner 3 and liner 4. The OD demands for different liner carriers are derived from real data (with modification due to the commercial confidentiality). In order to obtain the average transportation cost kij for shipping one TEU empty container from port i to port j, we mainly consider the bunker cost of a 5000-TEU ship with a common sailing speed (e.g., 20 knots), since the bunker cost is a 3000 ¼ 0:03), major component of the ship operating cost. Following Brouer et al. (2014), we assume kij ¼ 0:03 Disij (i.e., 205000 where Disij is the distance between port i and port j. 4.2. Results analysis Firstly, we investigate the benefit of liner carrier coordination. For simplicity, we show the results for two liner carriers. In order to explore the benefit of liner carrier coordination, two cases, i.e., case I (two liner carriers do not coordinate) and case II (two liner carriers coordinate) are mainly investigated. In case I, empty containers are not exchanged between two liner carriers. In this case, constraints (3) and (4) of the model P can be rewritten as follows:
X
k xkk ij ¼ ni ;
8 i 2 Sk ; 8 k 2 L
ð28Þ
j2Wk
X k xkk ij ¼ nj ;
8 j 2 Wk ; 8 k 2 L
ð29Þ
i2Sk
As shown in Table 1, it is clear that, in case I (i.e., two liner carriers do not coordinate), the total transportation costs for two liner carriers are 1.64 million USD and 1.38 million USD, respectively. When two liner carriers coordinate (i.e., case II),
Fig. 1. Ports in an Asia–Europe–Oceania shipping service network.
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the total transportation costs for two liner carriers become 1.18 million USD and 1.69 million USD, respectively. The overall transportation cost for two liner carriers can be reduced by 0.15 (i.e., (1.64 + 1.38) (1.18 + 1.69) = 0.15) million USD when they coordinate. Table 1 also shows the total empty container exchange cost paid to liner 1 and liner 2, respectively. We can find that liner 1 should pay 0.36 (i.e., 1.88 1.52 = 0.36) million USD to liner 2. Then, in case II, the total costs for liner 1 and liner 2 become 1.54 million USD and 1.33 million USD, respectively. Evidently, two liner carriers can benefit from the coordination with each other. Next, we show the results for the empty container exchange cost at each deficit port and the unit profit for delivering empty containers from each surplus port, as shown in Tables 2 and 3, respectively. In addition, the results for empty container allocation for the surplus ports of two liner carriers are shown in Tables A1 and A2, provided in Appendix A. The results for empty container allocation together with the empty container exchange costs can be used to verify the unit profits at any surplus port for delivering empty containers to different deficit ports. Table 2 shows the results for the empty container exchange costs at the deficit ports. As compared with other deficit ports, Bremerhaven has the lowest value of empty container exchange cost (i.e., 4.56 USD), and Tokyo has the largest value of empty container exchange cost (i.e., 339.87 USD). Moreover, the empty container exchange costs at the deficit ports in Europe are lower than 15 USD, since many empty containers are accumulated at the surplus ports in Europe, because of the imbalance between imports and exports for Asia-Europe trade. According to Theorem 2, we have proved that the unit profits at any particular surplus port of a liner carrier for serving different deficit ports of other liner carriers are identical. Here, we will further show that the unit profits for delivering empty containers from any particular surplus port of a single liner carrier to different deficit ports of this liner carrier are also identical. As shown in Table A2, Melbourne, Chittagong, Aqabah, Port Klang, Salalah and Jeddah are the surplus ports of liner 2, all of which only serve the deficit ports of liner 2. For instance, Brisbane and Sydney are served by Melbourne, and the transportation distances from Melbourne to Brisbane and Sydney are 1097 nautical miles and 582 nautical miles, respectively. According to Table 2, the unit profit for delivering empty containers from Melbourne to Brisbane is 260.91 USD (i.e., 293.82 0.03 1097 = 260.91), and the unit profit for delivering empty containers from Melbourne to Sydney is also P k 260.91 USD (i.e., 278.37 0.03 582 = 260.91). It seems that we also have gkj ¼ 0 when i2Sk xkk ij ¼ nj (8j 2 Wk ) is satisfied, P kk k xij ¼ nj (8j 2 Wk ) is satisfied, the empty consimilar to the third complementary slackness condition. Actually, when i2Sk tainer exchange cost at deficit port j of liner carrier k, which is dependent on gkj , is paid to liner carrier k herself. In other words, various values of gkj do not impact the coordination and interaction among different liner carriers. In order to minimize the objective function of INVP (19), gkj ¼ 0 can be obtained. Furthermore, when any particular surplus port of a single liner carrier k serves different deficit ports of different liner carriers including liner carrier k, one can find that the unit profits at this surplus port for delivering empty containers to different deficit ports are also identical. Actually, we obtain a
Table 1 Comparison between two cases. Total transportation cost (million USD)
Case I Case II
Total empty container exchange cost (million USD)
Liner 1
Liner 2
To liner 1
To liner 2
1.64 1.18
1.38 1.69
– 1.52
– 1.88
Table 2 Empty container exchange costs at the deficit ports. Deficit ports
Empty container exchange costs (USD)
Deficit ports
Empty container exchange costs (USD)
Brisbane Fremantle Adelaide Sydney Zeebrugge Le Havre Bremerhaven Hong Kong Cochin Chennai Nhava Sheve Jakarta Tokyo Nagoya Yokohama Kobe Pusan Kwangyang
293.82 205.2 245.49 278.37 9.57 14.7 4.56 308.52 210.78 216.42 204.78 258.09 339.87 333.09 339.51 328.92 319.41 320.67
Port Klang Dalian Xingang Qingdao Ningbo Shanghai Karachi Manila Singapore Yantian Xiamen Chiwan Colombo Kaohsiung Leam Chabang Jebel Ali Thamesport Ho Chi Minh
258.42 317.64 323.64 322.2 326.13 334.17 207.78 304.95 264.72 309.72 317.13 309.72 218.34 317.73 278.22 186.45 12.03 284.1
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Table 3 The unit profits for the surplus ports. Surplus ports
The unit profits (USD)
Surplus ports
The unit profits (USD)
Adelaide Antwerp Aqabah Brisbane Chennai Chittagong Colombo Dalian Fremantle Hamburg Hong Kong Jakarta Jebel Ali Jeddah Karachi Kobe
245.49 7.44 118.32 293.82 216.42 219.09 218.34 317.64 205.2 0 308.52 258.09 186.45 135.54 207.78 328.92
Kwangyang Le Havre Manila Melbourne Nagoya Nhava Sheve Port Klang Pusan Qingdao Rotterdam Salalah Singapore Sokhna Southampton Sydney Zeebrugge
320.67 14.7 304.95 260.91 333.09 204.78 258.42 319.41 322.2 7.59 171.78 264.72 97.32 15.15 278.37 9.57
unique profit for any surplus port when delivering one empty container to any possible deficit port. In Table 3, we show the unit profit at each surplus port for providing empty containers. Clearly, Hamburg has the lowest value of the unit profit, which is equal to 0 USD, and Nagoya has the largest value of the unit profit (i.e., 333.09 USD). Compared with other surplus ports, Hamburg is the farthest surplus port for delivering empty containers to the deficit ports, as shown in Fig. 1. Compared with other deficit ports, the deficit ports in Japan (e.g., Tokyo, Yokohama, etc.) have a large value of the empty container exchange cost, as shown in Table 2. Hence, as the nearest surplus port to Yokohama, Nagoya can obtain the largest value of the unit profit for delivering empty containers to Yokohama. In practice, when the empty container supply does not meet the demand, some liner carriers may lease empty containers. As compared with the container leasing prices provided by the container leasing companies, the perceived values of empty container (i.e., the unit profits and the empty container exchange costs) at different ports may have an impact on pricing in the container leasing market. For any particular surplus port, liner carriers would like to reposition empty containers if the unit profit at this port is larger than the off-leasing price. Otherwise, liner carriers would lease off their empty containers in order to avoid the repositioning costs. For any particular deficit port, liner carriers would like to lease in empty containers if the empty container exchange cost at this deficit port is larger than the leasing-in price. Otherwise, the coordination among liner carriers for exchanging empty containers would be favorable. Now, we investigate the stability of liner carrier coordination on empty container allocation by considering the coordination among different numbers of liner carriers. Table 4 shows the results of all combinations for the coordination among four liner carriers. The second column in Table 4 shows the different coordination cases. For example, (1, 2, 3) means that liner 1, liner 2 and liner 3 coordinate with each other. In order for easy comparison, Table 4 also shows the individual transportation cost in the non-coordination case. Evidently, one can find that any two of four carriers can benefit from the coordination with each other, as compared with the non-coordination case. In other words, the coordination formation is stable when the number of liner carriers is 2. When the number of liner carriers is 3 or 4, the coordination formation among liner carriers is also quite stable. This is because there is no motivation for some liner carriers to collude, in order to decrease their costs. In addition, results for cost allocation by using the Shapley value are also shown in Table 4. Clearly, our results are slightly different with the Shapley value. Table 4 Results for the coordination among different numbers of liner carriers. Total cost (million USD) Liner 1
Liner 2
Liner 3
Liner 4
Two carriers
(1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4)
1.54 1.63 1.63 – – –
1.33 – – 1.34 1.31 –
– 1.29 – 1.19 – 1.29
– – 1.92 – 1.82 1.92
Three carriers
(1, 2, 3) (1, 2, 4) (1, 3, 4) (2, 3, 4)
1.55 1.58 1.37 –
1.31 1.27 – 1.30
1.21 – 1.57 1.22
– 1.85 1.80 1.83
Four carriers Non-coordination Shapley value
(1, 2, 3, 4)
1.62 1.64 1.5
1.16 1.38 1.15
1.27 1.80 1.33
1.90 2.50 1.97
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5. Conclusions This paper has investigated an empty container allocation problem considering the coordination among liner carriers for exchanging empty containers. We further measure the perceived values of empty container at different ports. To solve our problems, we propose a two-stage optimization method. Numerical implementations show the following conclusions: (i) liner carriers can benefit from the coordination with each other; (ii) empty containers are delivered if found profitable; (iii) the coordination formation among liner carriers is found to be stable; (iv) the perceived values of empty container at different ports are measured. Furthermore, by comparing the perceived values of empty container at different ports with the container leasing prices provided by the container leasing companies, we hope our work is helpful for pricing in the container leasing market. As shown in our assumptions, there are some limitations in our work: (i) travel times for delivering empty containers are neglected; (ii) the weekly container demands are fixed and given in advance; (iii) we only consider the average transportation cost when repositioning empty containers; (iv) each liner carrier is assumed to have sufficient space for repositioning empty containers. There are some research issues we will investigate in the future. Firstly, we will extend our work by considering the weekly dependent container demands, since the container demands vary from week to week. Secondly, we will study the repositioning of empty containers and the routing of laden containers in a liner shipping network, by considering ship capacity constraints, ship operating cost, bunker cost, inventory cost and container handling cost, etc. Thirdly, we will investigate the impact of the container leasing prices on our results, in order to better understand the container leasing market. Acknowledgements We would like to thank anonymous referees for their useful comments, which significantly improve the presentation of the paper. This research is supported by the National Basic Research Program of China (2012CB725400), the National Natural Table A1 Empty container allocation for the surplus ports of liner 1. Surplus ports
Surplus containers
Brisbane
87
Adelaide
192
Melbourne
202
Sydney
191
Chittagong
112
Zeebrugge
107
Sokhna Hamburg Nhava Sheve
1433 527 96
Jakarta
296
Aqabah
521
Rotterdam
280
Salalah
381
Karachi
121
Manila
66
Jeddah
726
Singapore
146
Colombo
126
Jebel Ali
385
Southampton
1685
Liner carriers Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
Deficit ports with delivered empty containers Brisbane (87) Adelaide (192) Brisbane (202) Sydney (191) Ho Chi Minh (112) Zeebrugge (85), Manila (22) Ningbo (386), Shanghai (857) Yantian (190) Bremerhaven (24), Ho Chi Minh (52) Bremerhaven (148), Thamesport (303) Nhava Sheve (96) Jakarta (155), Manila (141) Yokohama (521) Hong Kong (59) Tokyo (221) Cochin (4), Nhava Sheve (229), Karachi (148) Karachi (121) Manila (66) Cochin (726) Xiamen (146) Colombo (126) Karachi (204), Jebel Ali (181) Xiamen (295) Ningbo (1390)
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Table A2 Empty container allocation for the surplus ports of liner 2. Surplus ports
Surplus containers
Liner carriers
Deficit ports with delivered empty containers
Fremantle
1296
Fremantle (19) Adelaide (475), Sydney (436), Jakarta (366)
Melbourne
821
Chittagong
51
Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner Liner
Antwerp Sokhna
923 1796
Le Havre
800
Hamburg
880
Hong Kong
56
Chennai
551
Nagoya
114
Kobe
246
Aqabah
284
Pusan
290
Kwangyang
856
Port Klang
174
Rotterdam
594
Dalian
1531
Qingdao
1431
Salalah
361
Jeddah
275
Southampton
1435
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
Brisbane (547), Sydney (274) Manila (25), Ho Chi Minh (26) Cochin (18), Kaohsiung (224), Leam Chabang (160), Ho Chi Minh (134) Manila (59), Yantian (76), Colombo (132), Kaohsiung (120) Port Klang (41), Ningbo (431), Chiwan (398) Singapore (543), Leam Chabang (383) Le Havre (5), Chiwan (626) Xiamen (169) Thamesport (4) Bremerhaven (876) Chiwan (56) Chennai (22), Yokohama (20), Port Klang (221) Chiwan (288) Nagoya (48), Yokohama (46) Yokohama (20) Yokohama (41), Kobe (162) Tokyo (43) Yokohama (284) Pusan (224) Shanghai (66) Kwangyang (249) Shanghai (607) Singapore (174) Hong Kong (25), Xiamen (569) Dalian (59), Xingang (95), Shanghai (812) Xingang (565) Qingdao (201), Shanghai (90) Shanghai (1140) Cochin (361) Nhava Sheve (275) Hong Kong (419), Tokyo (5), Yantian (583) Karachi (428)
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