This paper investigates the problem of containership sailing speed and fleet deployment optimization in an intercontinental liner shipping network. Under the consideration of the time value of container cargo, three kinds of impact of sailing speed changes on long legs of each liner route are analysed, and a time-based freight rate strategy is proposed. Then, the optimization problem is formulated as a mixed-integer nonlinear programming. Its goal is to maximize the total profits of a container liner shipping. To find the optimal solution to the model and improve the efficiency of model solution, a discretization algorithm is proposed. Numerical results verify the applicability of the proposed model and the efficiency of the algorithm. In addition, the time-based freight rate strategy is able to achieve more profit compared to a fixed freight rate strategy.
Container liner shipping services play an important role in maritime freight transportation [
Container shipping lines usually face two types of customer demands: long-term contractual demand and spot market demand [
Fleet deployment is one of the most important problems which container shipping lines have to face. Fleet deployment is to determine the number and type of ships to be assigned to the shipping routes [
Previous studies on the problem of SSFD assumed that containerships sail at fixed sailing speeds. Then, the problem of SSFD was simplified as the fleet deployment problem. The fleet deployment problem was first addressed in the literature by Perakis and Jaramillo [
Due to high bunker prices caused by high sailing speed, container shipping lines began to adjust the sailing speed to reduce the total operating cost [
These studies have significantly contributed to the development of mathematical programming for SSFD optimizations. However, to the best of the authors’ knowledge, the problem of SSFD optimization under TDFRS is still open.
The objective of this paper is to develop a model to achieve the optimal sailing speed of containership and the optimal number of deployed containerships under TDFRS with the goal of maximizing the total profit of a container shipping liner.
The contributions of this paper are threefold. First, it takes the initiative to address the SSFD problem under TDFRS while considering the time value of container cargo for an intercontinental liner network. This provides a reference for container shipping lines to design an optimal TDFRS and a freight rate for spot market customers. Consequently, the container cargo’s time value is converted into revenue, which increases the total profit of the container shipping liner. Second, the SSFD problem under TDFRS is an extension to the SSFD problem in the literature [
The remainder of this paper is organized as follows. Section
Consider a container shipping liner that operates a set of intercontinental routes which regularly serves a group of calling ports. In practice, the calling ports on each liner route form a loop and the sequences of these calling ports are determined in advance. A string of homogeneous containerships is deployed on each liner route to provide service once a week. The SSFD problem faced by the container shipping liner is how to decide the sailing speed of the containership and fleet deployment with the consideration of time value of container cargo within a planning horizon. The length of the planning horizon is assumed to be 3–6 months, as it is the maximum period of time over which the cost parameters can be regarded as unchanged [
Table
East Asia-U.S. West coast liner shipping network.
Route | Containership type | Calling ports |
---|---|---|
1 | 1 | Lianyungang ⟶ Shanghai ⟶ Ningbo ⟶ Long Beach ⟶ Seattle ⟶ Lianyungang |
2 | 2 | Tokyo ⟶ Qingdao ⟶ Shanghai ⟶ Ningbo ⟶ Los Angeles ⟶ Oakland ⟶ Tokyo |
3 | 2 | Qingdao ⟶ Shanghai ⟶ Ningbo ⟶ Los Angeles ⟶ Oakland ⟶ Qingdao |
4 | 1 | Taipei ⟶ Xiamen ⟶ Shekou ⟶ Yantian ⟶ Los Angeles ⟶ Oakland ⟶ Taipei |
It can be seen from Table
Under the consideration of the time value of container cargo, it is necessary to reoptimize sailing speed of containerships on long legs because it has three kinds of impact on SSFD optimization. First, it affects containerships’ sailing time on long legs, which is an important part of the transit time of container cargoes between corresponding port pairs. Meanwhile, inventory cost associated with container cargoes changes, especially for shippers with high container cargo values. If shippers pay part of the saved inventory cost to the container shipping liner, the container shipping liner achieves an increased revenue generated by the transit time savings. Second, it influences the number of deployed containerships on each liner route because the number of deployed containerships needs to be changed according to containerships’ sailing speed to maintain the weekly service frequency. Thus, a container shipping liner has to choose “fast steaming with less containerships” or “slow steaming with more containerships.” Different choices result in different containership cost. Third, it significantly affects the bunker fuel cost on long legs due to the relationship between bunker fuel consumption and sailing speed. Meanwhile, for short legs on each liner route, the impact of the reoptimizing sailing speed is negligible because of short oceanic distance [
The objective of the proposed optimization model is to reoptimize the sailing speed on each long leg on each liner route and the number of deployed containerships as well as the freight rate for the spot market customers. Before this, the optimal sailing speeds on each leg of each liner route need to be determined first without the consideration of the time value of the container cargo. The proposed models aim at maximizing the total profit. The total profit is calculated by the difference between the freight revenue and the total operating cost consisting of the bunker fuel cost and containership cost.
The notions used in this paper are introduced as follows: Sets Parameters Variables
According to the problem statement, the change of sailing speed has three kinds of impact on fleet deployment decisions considering the time value of the container cargo. The first impact is that it affects the transit time of containerships on the long legs on each liner route. If a container shipping liner increases sailing speeds of containerships, sailing time between port pairs can be shorten and the inventory cost of customers can be saved [
The inventory (holding) cost per unit is a linear function of time in storage [
It should be pointed out that when
The second impact is that it influences the number of deployed containerships, which is directly related to the voyage time on each liner route. The voyage time consists of the port calling time and the sailing time. To maintain a weekly service frequency, the following constraint should be satisfied:
The third impact is that it affects bunker fuel cost according to the relationship between bunker fuel consumption and sailing speed [
Similarly, the daily bunker fuel consumption on the long leg
The daily bunker fuel consumption on the long leg
Thus, the bunker fuel cost on each voyage
Based on the consideration of the time value of container cargo, the SSFD problem under the TDFRS can be formulated as the following mixed-integer nonlinear programming model [M1]:
Objective function (
Model [M1] is a mixed-integer nonlinear programming model with nonlinear terms in its objective function (
It should be pointed out that one of the most important pieces of model [M1] is the optimal sailing speeds
In order to get the optimal solution, a discretization algorithm is designed to transform model [M1] into a mixed-integer one. Then, it can be solved by the linear optimization solvers. The steps of the discretization algorithm are described as follows: without the consideration of the time value of the container cargo, a new mathematical programming is formulated based on a fixed freight rate strategy (FFRS). That is, freight rates for spot market customers between port pairs do not change as sailing speed changes. Therefore, the sailing speeds linearize constraints (
s.t.
All the constraints of model [M3] become linear constraints. Since the objective function (
The objective function (
Let
Since
According to equation (
Therefore, discretize the definition domain
Considering the fact that the sailing speed is usually taken to the decimal point after the unit of knots, we define
Therefore, the feasible domain of sailing speed can be discretely divided into
Correspondingly, the definition domain of
Note that
The optimal sailing speed
Thus,
Then, model [M3] is equivalent to the following model [M4]: obtain the optimal sailing speed considering the time value of the container cargo, the mathematical programming is formulated under the TDFRS. Then, this formulated programming can be transformed into an integer linear programming following the methods of variable substitution and piecewise linearization in Steps 2 and 3. The transformation process is shown in the appendix:
Constraints ( reoptimize the optimal sailing speed on long legas (
To summarize, the optimization results of the sailing speeds on each leg on each liner route and the number of deployed containerships under the FFRS can be obtained through Steps 1 to 4, and the optimization results under the TDFRS can be obtained through Steps 1 to 6.
Discretization of the new decision variables.
To evaluate the applicability of the proposed model and the efficiency of the designed algorithm, a real-case example provided by COSCO Shipping Liner Co., Ltd is used in this experiment. In this example, the interested intercontinental liner network consists of four Asia-Southwest America liner routes, as shown in Table
Port calling time of each port on each liner route.
Routes | Port calling time (hr) | |||||
---|---|---|---|---|---|---|
Port 1 | Port 2 | Port 3 | Port 4 | Port 5 | Port 6 | |
1 | 20 | 48 | 20 | 83.5 | 35 | — |
2 | 8 | 28 | 36 | 36 | 72 | 35.5 |
3 | 22 | 22 | 20 | 61 | 24 | — |
4 | 16 | 18 | 12 | 29 | 46.5 | 22.5 |
Oceanic distance of each leg on each liner route.
Routes | Distance (nautical miles) | |||||
---|---|---|---|---|---|---|
Leg 1 | Leg 2 | Leg 3 | Leg 4 | Leg 5 | Leg 6 | |
1 | 356 | 235 | 5 761 | 1 148 | 5 122 | — |
2 | 1 110 | 375 | 235 | 5 758 | 369 | 4 560 |
3 | 375 | 235 | 5 758 | 369 | 5 413 | — |
4 | 197 | 327 | 85 | 6 356 | 369 | 5 635 |
There are two types of containerships: 10060 TEUs and 8 400 TEUs, as shown in Table
Containerships.
Containership type | Capacity (TEUs) | Total owned containerships | Containership cost (USD/week) | Minimum speed (knots) | Maximum speed (knots) | Designed speed (knots) | Fuel consumption at designed speed (tons/day) |
---|---|---|---|---|---|---|---|
1 | 10 060 | 12 | 269 500 | 18.0 | 28.0 | 23.0 | 222.9 |
2 | 8 400 | 13 | 245 000 | 18.0 | 28.0 | 22.5 | 208.4 |
The other parameters required by the proposed model are set as follows. The bunker fuel price is 500 USD per ton according to the bunker market situation. The coefficient of the time value of container cargo is 8%, and the unit container cargo value on each liner route is set to be 225 000 USD [
After the above required parameters are determined, the mixed-integer nonlinear programming proposed in Section
The optimization results under different freight rate strategies (FFRS and TDFRS) and different revenue sharing rates (
Results obtained by the proposed optimization model under FFRS and TDFRS.
Freight rate strategy | Routes | Sailing speed (knots) | Number of containerships | Voyage time (hr) | Profit of each route (USD) | Total profit (USD) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Leg 1 | Leg 2 | Leg 3 | Leg 4 | Leg 5 | Leg 6 | |||||||
FFRS | 1 | 19.8 | 19.9 | 19.7 | — | 5 | 840 | 10 949 979 | 49 934 590 | |||
2 | 19.7 | 19.5 | 19.6 | 19.7 | 5 | 840 | 10 161 789 | |||||
3 | 18.0 | 19.5 | 18.0 | — | 5 | 803 | 11 209 699 | |||||
4 | 18.2 | 18.2 | 18.5 | 18.3 | 5 | 840 | 17 613 123 | |||||
TDFRS | 1 | 19.8 | 19.9 | 19.7 | — | 5 | 840 | 10 950 651 | 49 948 395 | |||
2 | 19.7 | 19.5 | 19.6 | 19.7 | 5 | 840 | 10 162 840 | |||||
3 | 18.0 | 19.5 | 18.0 | — | 5 | 803 | 11 209 699 | |||||
4 | 18.2 | 18.2 | 18.5 | 18.3 | 5 | 811 | 17 625 205 | |||||
1 | 19.8 | 19.9 | 19.7 | — | 5 | 745 | 11 202 298 | 51 697 312 | ||||
2 | 19.7 | 19.5 | 19.6 | 19.7 | 5 | 777 | 10 271 035 | |||||
3 | 18.0 | 19.5 | 18.0 | — | 4 | 672 | 11 745 238 | |||||
4 | 18.2 | 18.2 | 18.5 | 18.3 | 4 | 671 | 18 478 741 |
It can be seen from Table
When the revenue sharing rate is 0.5, the number of containerships on the four routes remains constant, which is different from the optimization results under FFRS. The variation of the sailing speed on the two long legs on each route is different. Specifically, the long leg sailing speed in the head-haul direction of routes 1 and 2 increase while those in the back-haul direction decrease. The sailing speed on the two long legs of route 3 remains unchanged, and the sailing speed on the two long legs of route 4 increases. It can be explained that the variations in the sailing speed on long legs affect the sailing time and consequently affect the total freight revenue. Meanwhile, the sailing speed affects the bunker fuel cost. Hence, to determine the optimal sailing speed on the long legs on each liner route, the container shipping liner has to make the trade-off between the total freight revenue and bunker fuel cost. If the average container demand in the head-haul direction (or the back-haul direction) is high, then the added value of the freight revenue is higher than the added value of the bunker fuel cost, and the container shipping liner increases its sailing speed on long legs (such as long legs in the head-haul direction of routes 1 and 2, and the long legs of route 4). If the average container demand in the head-haul direction (or the back-haul direction) is low, then the reduction of the total freight revenue is lower than the added value of bunker fuel cost, and the container shipping liner increases its sailing speed on long legs (such as long legs in the back-haul direction of routes 1 and 2). If the average container demand in the head-haul direction (or the back-haul direction) changes in a certain range, then the total profit is constant when the variations of the sailing speed is ignored, and the containerships remain the original sailing speed on long legs (such as long legs on route 3).
When the revenue sharing rate increases from 0.5 to 1, all the sailing speeds on the long legs of the four routes increase. With regard to the fleet deployment, the number of containerships on routes 1 and 2 remain constant, and the number of containerships on routes 3 and 4 reduce from 5 to 4. The reason is that, although increasing sailing speed on the long legs on each liner route results in the increase of bunker fuel cost, the total freight revenue obtained by the container shipping liner has more increase than the bunker fuel does. Moreover, the increase of sailing speed may lead to the reduction of the number of containerships and consequently reduce the containership cost. As a result, the increase in total freight revenue caused by increased sailing speed is higher than that of total operating cost caused by increased sailing speed. Therefore, the container shipping liner is expected to increase the sailing speed on long legs, and when the sailing speed of some routes increases to a certain value, the number of containerships decreases.
In addition, it can be seen from Table
Both the revenue-sharing rate and bunker fuel price have significant impact on the total profit achieved by the container shipper liner. To explore the relationship among the two factors and the sailing speed of long legs, the impact of the revenue sharing rate and bunker fuel price on the results obtained by the proposed optimization model under TDFRS is investigated. The bunker fuel price is ranged from 300 $/ton to 800 $/ton at 50 $/ton interval, and the revenue-sharing rate is ranged from 0.1 to 1.0 at a 0.1 interval. Sensitivity analysis results of the long legs’ sailing speeds on the four routes are obtained. Considering the space limitations, only the optimal sailing speed on the long legs in the head-haul direction (leg 3) and in the back-haul direction (leg 5) of route 1 are demonstrated in this paper, as shown in Figure
Sensitivity of the optimal sailing speed of route 1: (a) leg 3 and (b) leg 5.
It can be seen from Figure
Generally, the increase of the bunker fuel price will result in the decrease of the sailing speed, while the increase in the revenue sharing rate will result in the growth of the sailing speed. Nevertheless, the above sensitivity analysis results of the long leg sailing speed are not completely consistent with the existing knowledge. The reason is that the SSFD problem under the TDFRS obtains optimization results under the full consideration of the total freight revenue, bunker fuel cost, and containership cost. The sailing speed changes in the head-haul direction and those in the back-haul direction have impact on each other.
Figure
Changes of the total profit under the two freight rate strategies.
It can be seen from Figure
In this study, the SSFD problem in an intercontinental liner network with the consideration of the time value of the container cargo is investigated. The problem is first formulated as a mixed-integer nonlinear programming under the TDFRS. In consideration of the nonlinearity of the model, the piecewise linearization algorithm is designed to transform the model into an integer linear programming. The proposed model and the algorithm are evaluated by numerical examples. The results show that, considering the time value of the container cargo in the SSFD problem affects containerships sailing speed on long legs and the number of deployed containerships. Moreover, when the TDFRS is adopted for spot market customers, the optimization results obtained by the proposed model are able to not only increase the total profit for the container shipping liner but also provide a satisfactory level of service for customers.
It should be noted that the container demand between port pairs may fluctuate as freight rate changes. Therefore, the SSFD problem considering both the time value of the container cargo and the dynamic container demand should be explored in future research.
First, model [M1] linearization: the objective function (
Define
All the constraints in model [M3′] are linear, but the third term of the objective function contains nonlinear functions of
According to Theorem
Second, discretize the definition domain
Then,
Finally, the mixed-integer nonlinear programming model [M3′] can be transformed into an equivalent integer linear programming model [M5].
All the data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This study was supported by the National Natural Science Foundation of China (NSFC) (Grant no. 71372088).