Keywords

1 Introduction

The liner company needs to meet the most favorable basic conditions both technically and economically. Since the 1990s, scholars at home and abroad have been exploring route allocation and fleet planning for large liner fleets, seeking to establish a set of scientific mathematical methods and techniques to solve this problem. Perakis and Jaramillo were the first to define the route allocation problem, and they proposed a preliminary model and available solutions, which to some extent provided the basis for subsequent research has laid the foundation (Jaramillo 1991). The works of Wu Changzhong and Yang Qiuping are the earlier works on the route allocation problem in China, which established a linear model with operating cost and opportunity cost as the objective function and capacity and volume limitation as the constraints from the perspective of shipping companies, which laid a good foundation for the subsequent studies (Wu 1992; Yang 2011). XIE X and others proposed a nonlinear model for fleet planning (Xinlian 1993). Zhao Gang explored a more realistic nonlinear impact of route allocation (Zhao 1997). The studies of An Fen, Huang Yong have greater significance for the research of this paper, and these are more targeted studies for the route optimization of inland river container liner system in China (An 2014; Huang2007). Huang Yong initially studied the optimization of inland river container routes in Yangtze River, and achieved the lowest total cost of the whole system through route search and model solution (Huang 2007). An Fen analyzed the container capacity of single vessel and the container loading capacity of each port of call on the basis of Huang Yong’s studies (An 2014). The empty container transfer problem caused by the imbalance of two-way transportation was also considered. By the 21st century, the study by Maxim A. Dulebenets extended the cooperation agreement between liner carriers and sea container terminal operators by considering, for example, the negotiation of vessel time windows (TW) and loading and unloading rates between liner carriers and sea container terminal operators (Dulebenets 2018; Dulebenets 2019). New forms of cooperative agreements will improve liner shipping and ocean container terminal operations. Chen et al. provide a systematic overview of research on liner shipping alliance management, including forming alliances, selecting partners, and designing cooperative mechanisms (Chen et al. 2022). This has similarities with Dulebenets’ research on alliances.

Mamedio noted that alliances can facilitate the effective integration of information and resources among members, and are a channel for sharing technology and other capabilities, so they can effectively integrate resources and capabilities among members (Mamedio 2019). Cariou and Cuillotreau’s study argues that alliances can be effective in addressing the problem of overcapacity, noting that alliances can play a greater role in reducing overcapacity when the number of competitors is limited. They point out that when the number of competitors is limited, alliances can play a greater role in reducing overcapacity (Cariou 2021).

Based on the above studies, there is an end to the study of alliances between inland shipping and fleet companies for liner shipping issues, from the initial maritime to inland waterways, and from the initial individual interests to the current alliance interests. The study in this paper is inspired by the above-mentioned studies, but focuses more on the impact of company alliances on benefits for particular vessel types, examining the overall fleet benefits under alliances and non-alliances. It also conducts a numerical analysis on the Han-Hai series of the 1,000-case fleet of the Han-Shen Line.

This study focuses on the ship scheduling of the Wuhan-Shanghai Hanshin line, and focuses on the 1,000 container fleet “Hanhai Series”. The five thousand container ships belong to three companies, among which Hanhai 2 and 3 are operated by Wuhan COSCO Shipping Container Transport Company Limited, Hanhai 7 is operated by Wuhan Changwei International Shipping Industry Co. The five vessels operate the direct river-sea service from Yangluo Port to Yangshan Port. The ports (in the order of visiting ports) through which the service passes include Yangluo Port, Maanshan Port, Jiujiang Port, Taicang Port, Yangshan Port and other series of ports. The containers transported have a corresponding starting port (Original Port) and destination port (Destination Port). Between each two ports can form a port pair (O,D). Linear arrangement of any two ports on the inland waterway trunk line may have a demand for container traffic between them, called OD traffic (An 2014). Han Shen Line container liner route network consists of several liner routes. Considering the alliance situation, each route may be operated by different carriers. However, its initial starting port is Yangluo port and destination port is Shanghai port, as shown in Fig. 1. The ports of call for each ship on a route are not necessarily the same for the outbound and return trips, as shown in Fig. 2.

2 Inland Container Liner Problem Description

This section describes in detail the liner shipping routes to be analyzed and presents the main modeling assumptions.

2.1 Network Planning Problem Description

This study focuses on the ship scheduling of the Wuhan-Shanghai Hanshin line, and focuses on the 1,000 container fleet “Hanhai Series”. The five thousand container ships belong to three companies, among which Hanhai 2 and 3 are operated by Wuhan COSCO Shipping Container Transport Company Limited, Hanhai 7 is operated by Wuhan Changwei International Shipping Industry Co. The five vessels operate the direct river-sea service from Yangluo Port to Yangshan Port. The ports (in the order of visiting ports) through which the service passes include Yangluo Port, Maanshan Port, Jiujiang Port, Taicang Port, Yangshan Port and other series of ports. The containers transported have a corresponding starting port (Original Port) and destination port (Destination Port). Between each two ports can form a port pair (O,D). Linear arrangement of any two ports on the inland waterway trunk line may have a demand for container traffic between them, called OD traffic. Han Shen Line container liner route network consists of several liner routes. Considering the alliance situation, each route may be operated by different carriers. However, its initial starting port is Yangluo port and destination port is Shanghai port, as shown in Fig. 1. The ports of call for each ship on a route are not necessarily the same for the outbound and return trips, as shown in Fig. 2.

Fig. 1.
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Liner alliance network

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Fig. 2.
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Route structure

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Under liner alliances, there is a need to consider the issue of slot chartering. Which refers to the act of liner companies chartering slots from cargo ship operators on one or more routes so as to operate the relevant routes without increasing capacity input, and paying slot chartering fees regularly. The space lessor can earn rental income and effectively solve the problem of surplus capacity due to declining demand.

Therefore, assuming that the Hanseatic series of operating companies reach an alliance, different companies can trade their own ships as resources by signing a charter agreement stipulating the basic amount of chartered space. At this point, the problem changes to the Hanseatic liner fleet with a certain scale, through the cargo source survey to grasp the amount of freight demand and the level of freight rates between any ports on its operating routes during the study period, and design the optimal alliance fleet plan to achieve the maximum overall return of the alliance. Each carrier can also maximize its own interests when operating according to the scheme of the maximum overall return of the alliance, so as to maintain the willingness of the alliance cooperation.

2.2 Network Planning Problem Assumptions

Since the route network planning of inland container liner transportation system involves numerous factors and details, if all the details are considered, the model is complicated and difficult to study. For the convenience of the study, the following hypotheses are proposed.

  1. (1)

    The study period is one year, one quarter, one month or other unit of time. The line is operating normally and there is no port leakage problem.

  2. (2)

    The volume of traffic between any two port pairs is deterministic and can be predicted from prior data, and the volume of traffic is essentially constant over a period of time; the level of tariffs has been determined, and the number of chartered slots and chartered tariffs in the liner alliance are also determined.

  3. (3)

    The container type is unique, and only TEU containers, i.e. 20-foot containers and only full container shipments are considered in this study, without considering the case of LCL shipments.

  4. (4)

    The optimal speed of each type of vessel on each route during the study period is predetermined; the time when the vessel enters and leaves the port is fixed and does not vary according to the type of vessel, and no congestion and delay problems occur.

  5. (5)

    The fixed costs of containers during the study period are not considered separately, i.e. the depreciation, repair, storage, insurance, chartering and other costs of containers are incorporated into the ship’s costs; at the same time, it is assumed that the fixed costs of ships are fixed and can be determined in advance according to the actual operation.

3 Inland Container Liner Mathematical Problem

3.1 Illustration of Symbols and Parameters

See Table 1.

Table 1. Nomenclature
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3.2 Modeling

3.2.1 Model Objective Function

To solve the above problems, a mathematical model of shipping space leasing and allocation is established to maximize the total fleet profit for Liner Companies during the study period, as in Eq. (1) (total revenue minus total operating cost)]

$$Max \, {\text{Z}}_{r}={P}_{r}-{S}_{r}-{C}_{r}$$
(1)

(1) Revenue of Ships at route r during planning period refers to the total profit of shipping containers transported by liner companies during the actual transportation of goods on THE OD port pair at route r which is affected by freight rate and volume, number of cabins rented and unit price of each cabin. Provided that it is assumed that freight rate and volume are known in advance, so the profit is expressed as in Eq. (2):

$${P}_{r}={\sum }_{o,d\in {N}^{*}}{p}_{o,d}^{r}{q}_{o,d}^{r}+{e}^{r}{f}^{r}$$
(2)

(2) Operation cost of Ships at route r during planning period refers to the total cost of shipping containers transported by liner companies during the actual transportation of goods which is affected by sailing distance, speed, time, number of attached ports, Berthing time in port fuel consumption and container loading influence. The operating cost of ships at route r in the planning period mainly contains of the fuel cost, including fuel, moisture and materials; port charges, including port charges for sluice and dam turning, which are not considered in this paper; as well as container handling fee. It is expressed as in Eq. (3)

$${S}_{r}=Fue{l}_{r}+Por{t}_{r}+Con{t}_{r}$$
(3)

Fuel consumption could be calculated as in Eq. (4):

$$Fue{l}_{r}={\sum }_{k}({\alpha }_{k}\cdot {T}_{k}^{r}\cdot N)$$
(4)

Berth cost at port p route r could be calculated as in Eq. (5):

$$Por{t}_{r}={\sum }_{k}{C}_{p}$$
(5)

Handling cost at port p route r could be calculated as in Eq. (6):

$$Loa{d}_{r}=cl{\sum }_{o\in N}{\sum }_{d\in N}{q}_{od}^{r}$$
(6)

It’s assumed that there is no LCL transport situation, so in this system all THE OD port pair for container loading and unloading expenses are only related to the total number of containers, namely, the number of loading and unloading is constant, so we can equal the container handling fee and the product of unit price and the amount of container. The container handling fee doesn’t change by optimization scheme. Thus, it could be treated as a constant value.

(3) Fixed cost of ships at route r during planning period

Fixed cost of ships at route r during planning period refers to ship once the leaves, is not affected by the container transport costs, including the crew wages, employee welfare, staff insurance premium, labor protection, ship maintenance fees, transportation and regulation fees, ship repair and other fees. For the simple model, we simply divided the fixed cost into two parts. It could be calculated as in Eq. (7):

$${C}_{r}={\sum }_{k}\frac{{T}_{k}^{r}\cdot N}{24}({H}_{k}+{Q}_{k})$$
(7)

At present, there is one variable to be processed. It contains round-trip sailing time and ship in-and-out time, which is calculated as in Eq. (8):

$${T}_{k}^{r}=T{N}_{k}^{r}+TE{O}_{k}^{r}$$
(8)

As for the sailing time, it could be calculated as in Eq. (9):

$$T{N}_{k}^{r}=\frac{{d}_{r}}{{S}_{rk}}$$
(9)

As for the ship in-and-out time, it could be calculated as in Eq. (10):

$$TE{O}_{k}^{r}={\sum }_{o\in N}{\sum }_{d\in N}({x}_{od}^{r}\cdot PT)$$
(10)

Combined with what has been stated above, the objective function can be integrated into

$$\eqalign{ & {Z_r} = {\sum _{o,d \in {N^*}}}p_{o,d}^rq_{o,d}^r + {e^r}{f^r} - {\sum _k}\left( {S_k^r \cdot {\alpha _k} \cdot T_k^r \cdot N} \right) \cr & - {\sum _k}{C_p} \cdot T_k^r - cl{\sum _{o \in N}}{\sum _{d \in N}}q_{od}^r - {\sum _k}\frac{{T_k^r \cdot N}}{{24}}({H_k} + {Q_k})) \cr}$$
(11)

3.2.2 Model Constraints

(1) Ship cargo carrying capacity constraint, the sum of all shipping Spaces starting from the same port O and those leased to alliance partners does not exceed the maximum carrying capacity of the ship:

$${\sum }_{d}^{D}{q}_{od}^{rm}+{f}^{r}\le {U}^{r}\forall o\in N$$
(12)

(2) Ship cargo carrying capacity constraint, the sum of all shipping Spaces terminating at the same D port does not exceed the sum of shipping Spaces leased from alliance partners and the maximum carrying capacity of ships:

$${\sum }_{o}^{O}{q}_{od}^{rm}\le {U}^{r}+{f}^{r}\forall d\in N$$
(13)

(3) Lease constraints (the number of slots rented to alliance partners on self-operated routes does not exceed the maximum shortage of alliance partners):

$$f\le {\beta }^{r}{z}^{r}$$
(14)

(4) Lease constraints (the number of slots rented from the alliance partner on other routes does not exceed the number of slots available for leasing by the Alliance partner)

$${f}^{r}\le (1-{\beta }^{r}){b}^{r}$$
(15)

(5) Variable constraints (variables are non-negative and integers)

$$f,{f}^{r},{q}_{od}^{r}\in N$$
(16)

4 Inland Container Liner Numerical Experiments

On condition that Wuhan Changhai international shipping agency co., LTD. (hereinafter referred to as “company A”), Wuhan Changwei international shipping industrial co., LTD. (hereinafter referred to as company B), Wuhan Chinese transportation maritime container transport co., LTD. (hereinafter referred to as “company C) The three companies have opened a total of 3 multi-port docking circular routes on both ends of Yangluo port, Shanghai port, a total of six port area of liner alliance. Company A operates R2, Company B R3, and Company C R4, while R1 is the direct route from Yangluo to Yangshan of Han-Shen Line. The route structure of the four routes involved is shown in the graph (Figs. 3, 4, 5 and 6).

Fig. 3.
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All the routes

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Fig. 4.
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R1

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Fig. 5.
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R2

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Fig. 6.
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R3

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In addition, the transport capacity of company A on the R1 route is 2280TEU. According to the initial alliance agreement, the maximum number of cabins that company A can rent in R2 is 500TEU, and the maximum number of cabins that company B and Company C can rent in R3 and R4 are both 500TEU. For the convenience of listing, the port names are replaced as shown in the Table 2:

Table 2. Port code description
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Due to the particularity of this route, the time from Yangluo port to Yangshan port is the optimal target, and port affiliation is almost not involved. Therefore, this study focuses on the return route affiliation. The research focus is on the shipping space leasing of inland river container liner. Provided that the known transport demand between ports, the research assumes that the transport volume between any two ports is determined, which is not the research content of this paper. Based on the assumption of the growth trend of container transport volume between ports returning from Han-Shen Line, the transport demand between ports operated by each company during the planning period (one week) is shown in the Table 3.

Table 3. Route related data.
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MATLAB was used to calculate and optimize respectively the profit of company A, B and C, while the other two companies were taken as constraints, and the mutual leasing results were obtained as shown in the Tables 4 and 5.

Table 4. Result of space exchange between routes
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Table 5. Results of seat allocation for each route of Company A
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According to the calculation, company A, B and C can obtain additional potential customers on other lines and increase their income after implementing the shipping space leasing of liner lines. On the basis of the model, variables and constraints related to mutual leasing are removed and the same method is used to calculate. Taking Company A as an example, its total revenue is reduced by 980,000 yuan. In order to verify the robustness of the model, on the basis of the above calculation example, two groups of different demand values are taken respectively, and other factors and data are controlled unchanged. The benefits are calculated and compared with the non-reciprocal leasing strategy, as shown in the Table 6:

Table 6. Comparison of benefits under different circumstances
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It can be seen from the table that adopting the mutual leasing strategy can bring more benefits to the shipping company through multiple groups of different demand values, which fully proves the robustness of the algorithm.

It can be seen from the above calculation results that the optimal model of space allocation for Wuhan thousand-container fleet of Han-Shen line based on alliance mutual leasing strategy constructed in this paper can not only expand the route coverage of the liner company, but also improve the space utilization rate of the liner company and bring additional operating benefits to the liner company. Therefore, the inter-leasing strategy of inland line liner alliance is a long-term win-win strategy of benefit sharing and risk sharing, and its optimal decision of inter-leasing and allocation is of vital importance to improve the market share and competitiveness of liner companies.

5 Conclusions

In the light of the characteristics of inland container liner shipping alliance space rent, each strategy is studied under the spandex optimization allocation problem of shipping space, the integer programming model is built to maximize the profit of the liner companies and example of Hanhai series is used to verify. Results of shipping space and rent and space distribution optimization decisions are got. It has certain reference value and guiding significance for liner alliance cooperation.

This study considers the space rent and space distribution of the decision, but only single optimal routes of a company operating decisions as the goal, while at this time of the other company are seen as a whole, without considering the influence of the individual. In the future study, we can build the model which considers the interests of the global optimization model from the perspective of the liner alliance as a whole in order to better realize the goal of win-win cooperation of liner alliance. In the meantime, this study ignores the difference of the two companies’ operating cycles and has not added the constraint of shipping cycle. Time cycle can be added in the future to get closer to the actual operation situation.