Towards Sustainable Shipping: Joint Optimization of Ship Speed and Bunkering Strategy Considering Ship Emissions

Abstract

Maritime regulators are closely monitoring the progression of green shipping, and liner companies are seeking strategies to meet tough ship emission rules. To reduce the operating cost while conforming to the increasingly strict environmental regulations, the study first constructs a mixed-integer nonlinear optimization model. Subsequently, the nonlinear parts in the objective function and constraints are transformed into linear forms. Thereafter, the model is applied to the Asia–Europe route of the CMA CGM Shipping Company to find the planned speeds and bunkering strategies for container liners sailing in expanded emission control areas (ECAs) that will be implemented in the future. Finally, a sensitivity analysis is performed to examine the influence of bunker tank capacity and fuel price difference on the operating cost, carbon dioxide emission, bunkering strategy and planned sailing speed. The study contributes to determining the optimal tank capacity and developing bunkering strategies at different fuel price differences. With stricter policies, operators must strategically choose refueling ports, adjust refueling amounts, and optimize planned sailing speeds based on ship and route data. The proposed approach provides a solution to the contradiction between compliance with environmental regulations and cost-effectiveness of shipping companies and is of great significance for promoting the sustainable development of the waterway transportation industry.

1. Introduction

In view of the environmental pollution caused by ships, the United Nations, the International Maritime Organization (IMO) and many countries have formulated strict emission standards and regulations, which mainly include the United Nations Framework Convention on Climate Change (UNFCCC), the Paris Agreement and the Kyoto Protocol, requiring developed countries to take responsibility for reducing the emission of air pollutants and greenhouse gases (GHGs) [1,2,3,4]. IMO has set a target for reducing GHGs from ships, requiring a 20–30% reduction in annual emissions by 2030 relative to the 2008 level [5]. Accordingly, starting from 1 January 2020, ships sailing outside Emission Control Areas (ECAs) are required to use fuel oils with a sulfur content no higher than 0.5% m/m [6]. In China, the sulfur content of fuel oils used in the coastal ECAs must not exceed 0.5% m/m, and even tighter regulations are in place for inland waterways [7]. Furthermore, the United Nations actively supports investment in green technologies for waterway transportation and continues to promote research and the practical application of low-carbon fuel oils with broad application prospects, such as ammonia, methanol, hydrogen energy and low-sulfur fuel oils [8,9]. Overall, by implementing these measures, including a comprehensive approach to technological innovation, international cooperation and effective regulation, the United Nations has accelerated the process of sustainable development in the shipping industry.

To comply with marine fuel oil sulfur content regulations, extensive research has been conducted to explore technologies and strategies for reducing ship emissions, including installing scrubbers, using liquefied natural gas (LNG) and switching to low-sulfur oils [10,11,12]. Different from the high cost of scrubbers and LNG, bunker switching technology is simple, making it a prevalent choice among shipping companies. However, compared to traditional fuel oils, the cost of switching to low-sulfur fuel oils is higher, accounting for a significant portion of ship operating expenses, which puts enormous economic pressure on ship operators [13]. Studies indicate that fuel consumption increases cubically with sailing speed [14], suggesting that optimizing a ship’s speed when entering or leaving an ECA can effectively reduce fuel cost. In this regard, many scholars have proposed various optimization models and algorithms to reduce fuel usage [15]. The ship’s sailing speed can be optimized to minimize the total cost by constructing various cost-minimization models [16,17,18]. These studies not only provide a variety of models for optimizing bunkering strategies but also help to provide ship operators with the most cost-effective solutions when faced with the high cost of low-sulfur fuel oils.

To date, academics have proposed a variety of methods to reduce carbon emissions from ships. The impact of carbon tax collection on the operation of ships and ports was investigated with the objective of minimizing the operating cost [19]. A carbon tax model was incorporated into a carbon trading system to identify the most cost-effective route [20]. The carbon emissions and operational costs for container ships were calculated, and the environmental and financial impacts on marine container transportation were assessed [21,22]. Studies have found that of the three fuel oils, namely marine gas oil (MGO), heavy fuel oil (HFO) and liquefied natural gas, HFO is associated with more significant economic advantages, while LNG has greater environmental benefits. These studies strive to strike a balance between economic profitability and environmental sustainability, all the while adhering to environmental policies.

Overall, these studies are based on traditional regulations, with the optimization of sailing routes, speeds and refueling strategies. However, after 1 May 2025, the Mediterranean ECA will come into effect [6], and the fuel sulfur limit within the ECA will be changed to 0.1% m/m. Meanwhile, the upper limit of sulfur content of fuel oils used outside ECA has been revised to 0.5% m/m, while most previous studies focused on the limitation of 3.5% m/m [23]. The upgrading of these policies has had a significant impact on the fuel cost of shipping companies, but there is limited research on the impact of these policies on ship operations.

To date, few studies have considered the economic and environmental impacts of the Mediterranean Sea ECA on shipping companies, and this study includes possible impacts. The rules for charging shipping companies are as follows: 50% of the actual emissions generated during navigation when departing or berthing at a port outside the European Union (EU), and 100% of the actual emissions generated during navigation and berthing between two EU ports [24].

This study constructs an optimization model for the liner service on the Asia–Europe shipping route, which quantifies the impacts of low-sulfur regulations as well as EU carbon emission policy on shipping companies, covering the quantification of total weekly operational cost, time delay penalty, carbon emission cost and bunker fuel cost. To reduce operating cost and comply with environmental regulations, this study incorporates the EU’s environmental policy, which charges shipping companies based on voyages, ports of call and the charge percentage for carbon dioxide emissions generated by ships. The liner company can obtain the optimal bunkering strategy and sailing speeds on different voyage segments with the help of the proposed model. Through the sensitivity analysis of bunker tank capacity and fuel price spread, this study identifies the optimal bunker tank capacity thresholds and bunkering strategies under different price spreads. The container liner data are real, and the strategies given are feasible.

The research is organized as follows: Section 1 illustrates the background and reviews the literature. A mixed-integer nonlinear optimization model is developed and linearized in Section 2. The data and results are shown in Section 3, followed by a discussion in Section 4. Conclusions and recommendations are given in Section 5. The study enables shipping companies to reduce operating costs while complying with environmental regulations, providing a reference for shipping companies to make more eco-friendly operational decisions.

2. Methods

2.1. Problem Description and Assumptions

This study investigates the transportation process of the liner service on the Asia–Europe route, which consists of n ports and passes through the EU region. The route starts and ends at the same port, creating a closed loop, as shown in Figure 1. The liner company provides shipping services under the premise of a sulfur restriction policy and faces fuel switching problems for each voyage segment, i.e., using fuel oils with a sulfur content of 0.1% m/m within the ECAs and 0.5% m/m outside the ECAs. Since lower-sulfur fuel oils are more expensive, ship managers often choose to slow down in ECAs to reduce fuel cost, but this increases travel time and affects service quality. To balance sailing speed and service level, this study introduces a soft time window to regulate arrival times at ports and imposes economic penalties for late arrivals. Furthermore, with limited bunker tank capacity and varying oil prices at different ports, ship operators urgently need to develop optimal bunkering strategies, which involve selecting bunkering ports and determining the refueling amount of fuel oils to minimize operating cost.

The model is developed based on the following assumptions:

(1)

The main engine uses fuel oil with a sulfur content of 0.1% m/m when the liner vessel is sailing within the ECAs and 0.5% m/m when sailing outside the ECAs. Auxiliary engines use MGO throughout the voyage and in port [26,27].

(2)

The port charges, port handling costs and weather conditions are not considered [15,16].

(3)

The fuel oil switching time is not considered, the port is free of congestion and ships do not have to queue or wait at the port [28].

(4)

The fuel consumption of boilers is not considered because it is low [29].

(5)

The vessel only refuels at the ports of call, and the fuel prices at port are known before the vessel departs. The frequency of the departures is weekly, and the total time for a round trip is a constant [18].

2.2. Model Formulation and Cost Analysis

2.2.1. Parameters and Variables

(1)

Sets and parameters

$N=\left\{\mathrm{1,2},...,n\right\}$: the set of ports on the route, where n is the total number of ports;

$Y$: the set of EU ports visited by the vessel;

$\mathcal{B}$: a set of sailing segments along the route, where in segment i, i represents the port of departure, and $(i+1)$ denotes the port of arrival;

$S_i^E$: distance traveled by the vessel within the ECAs of segment i, nm;

$S_i^N$: distance sailed outside the ECAs of segment i, nm;

$T:$ round trip time, hour;

$t_i^s$: dwell time at port i, including docking, loading and unloading time, hour;

$T_i^e,T_i^l$: permitted earliest/latest arrival time of a vessel at port i, hour;

$C_i^{MGO}$: fuel price of MGO at port i, USD/ton;

$C_i^{VLSFO}$: fuel price of VLSFO at port i, USD/ton;

$C_g$: the fixed operating cost for each refueling operation, USD;

$H^{MGO}$: the maximum capacity of MGO tanks, ton;

$H^{VLSFO}$: the maximum capacity of VLSFO tanks, ton;

$q_{max}$: the maximum number of refueling operations on a voyage;

$k_{max}$: the maximum number of vessels allocated to a voyage;

$\psi$: the penalty for violating time window constraint, USD/hour;

$\varphi$: the carbon price, USD/ton;

${\eta }_{co2}$: carbon emission factor;

$\lambda$: the proportion of carbon emissions included in the EU environmental policy every year, %;

$\mathsf{\omega }$: the actual load capacity, TEU;

${\omega }_0$: the rated load capacity, TEU;

$\mathsf{\alpha }$: the influence coefficient;

$C_E$: weekly operational cost of a vessel, USD/week;

$V_d$: the designed speed, knot;

$V_{min}$: lower limit of sailing speed, knot;

$V_{max}$: upper limit of sailing speed, knot;

$R_M$: fuel consumption rate of main engine, g/kWh;

$R_A$: fuel consumption rate of auxiliary engines, g/kWh;

$L_M$: load factor of main engine;

$L_A$: load factor of auxiliary engines;

$P_M$: the rated power of main engine, kW;

$P_A$: the rated power of auxiliary engines, kW;

$F_{\Omega }$: fuel consumption constant for ship’s main engine, ton/hour;

$F_{\delta }$: fuel consumption constant for ship’s auxiliary engines, ton/hour;

(2)

Variables

$F_i^M$: fuel consumption of main engine of a vessel sailing on segment i, ton;

$F_i^A$: fuel consumption of auxiliary engines on segment i, ton;

$F_i^P$: fuel consumption of auxiliary engines of a vessel dwelling at port i, ton;

$E$: the total actual emissions, ton;

$E_i$: actual carbon emissions from ships sailing on segment i and dwelling at port i, ton;

$E_i^s$: the actual carbon emissions of a vessel sailing on voyage segment i, ton;

$E_i^p$: the actual carbon emissions of a vessel dwelling at port i, ton;

${\mathrm{E}}^{\mathrm{E}\mathrm{U}}$: the total EU-related carbon emissions, ton;

$E_i^{EU-s}$: the EU-related carbon emissions from a vessel sailing on segment i, ton;

$E_i^{EU-p}$: the EU-related carbon emissions from a vessel dwelling at port i, ton;

$C_{ZE}$: total weekly operational cost of a vessel, USD;

$C_T$: the time delay penalty, USD;

$C_{\mathrm{A}}$: carbon emission cost, USD;

$C_F$: bunker fuel cost, USD;

$V_i^E,V_i^N$: sailing speed within/outside the ECAs on segment i, knot;

$M_i\left(V_i^E\right)$, $M_i\left(V_i^N\right)$: fuel consumption of main engine per unit distance for a vessel sailing within/outside the ECAs on segment i, ton/nm;

$k$: the number of vessels operating on the circular voyage;

$t_i^a$: arrival time at port i, hour;

$y_i^{1MGO}$,${ y}_i^{2MGO}$: stock of MGO when a vessel arrives at/leaves port i, ton;

$y_i^{1VLSFO},y_i^{2VLSFO}$: stock of VLSFO when a vessel arrives at/leaves port i, ton;

$w_i^{MGO}$, $w_i^{VLSFO}$: the bunkering amounts of MGO/VLSFO at port i, ton;

$x_i^{MGO}$: if a vessel bunkering MGO at port i, $x_i^{MGO}=1;$ otherwise, $x_i^{MGO}=0$;

$x_i^{VLSFO}$: if a vessel bunkering VLSFO at port i, $x_i^{VLSFO}=1;$ otherwise, $x_i^{VLSFO}=0$.

2.2.2. Operating Cost Analysis

• The total weekly operational cost, C_ZE

According to Refs. [19,21,22], the total weekly operational cost, C_ZE can be determined by

$$C_{ZE}=k{C}_E$$

(1)

2.

Time delay penalty cost, C_T

To reflect uncertainties in sea voyages, this study considers soft time windows instead of rigid ones. For port i, the arrival time window is [$T_i^e,T_i^l$] and the penalty cost for each unit of delay is $\psi$. According to Refs. [15,17], the total time delay penalty for the liner service, $C_T$, can be defined as follows:

$$C_T={\sum }_{i=1}^n\psi {[t_i^a-T_i^l]}^{+}$$

(2)

3.

Carbon emission cost, C_A

To cope with climate change [30,31], the EU has released a carbon emission policy; i.e., from 1 January 2024, carbon emission cost is levied on cargo and passenger ships of 5000 gross tons and above, with the collection rules as follows: (1) For voyages between EU and non-EU ports, i.e., voyages starting or ending at EU ports, the carbon emission cost is calculated based on 50% of the actual emissions generated during the navigation period. (2) For ships sailing within the EU region, the carbon emission cost is calculated based on 100% of the actual carbon emissions generated during the voyage. (3) For ships calling at EU ports, the carbon emission cost is based on 100% of the actual carbon emissions generated during the call [32].

According to the EU carbon emission policy, the carbon emission cost incurred in the previous year will be levied in the following year [21,22]. This study assumes that, unlike the EU rules, liner companies will optimize current sailing speed to reduce the operating cost based on the current year’s fuel consumption [14]. To calculate the carbon emission cost paid by liner companies, the calculation methods of fuel consumption, carbon emissions and carbon emission cost are introduced in turn.

(1)

Fuel consumption

When the vessel sails on segment i, the fuel consumption of the main engine, $F_i^M,$ is shown as

$$F_i^M=M_i\left(V_i^E\right)\times S_i^E+M_i\left(V_i^N\right)\times S_i^N$$

(3)

According to Ref. [14], the fuel consumption of the main engine is directly proportional to the third power of the sailing speed. Therefore, when a vessel sails within/without ECAs at speed $V_i^{E }\mathrm{o}\mathrm{r} V_i^N$, the amounts of fuel consumed by the main engine per unit distance, $M_i\left(V_i^E\right)$ and $M_i\left(V_i^N\right)$, are as follows:

$$M_i(V_i^E)=[\alpha +(1-\alpha ){\displaystyle \frac{\omega }{{\omega }_0}}][F_{\Omega }\times {\displaystyle \frac{{(V_i^E)}^2}{{{(V}_d)}^3}}]$$

(4)

$$M_i(V_i^N)=[\alpha +(1-\alpha ){\displaystyle \frac{\omega }{{\omega }_0}}][F_{\Omega }\times {\displaystyle \frac{{(V_i^N)}^2}{{{(V}_d)}^3}}]$$

(5)

$$F_{\Omega }=\left(R_M{P}_M{L}_M\right)\times {\displaystyle \frac{1}{{10}^6}}$$

(6)

When the vessel sails on segment i, the fuel consumption of the auxiliary engines [21], $F_i^A,$ is as follows:

$$F_i^A=F_{\delta }\times \left({\displaystyle \frac{S_i^E}{V_i^E}}+{\displaystyle \frac{S_i^N}{V_i^N}}\right)$$

(7)

$$F_{\delta }=\left(R_A{P}_A{L}_A\right)\times {\displaystyle \frac{1}{{10}^6}}$$

(8)

When the vessel dwells at port i, the fuel consumption of the auxiliary engines, $F_i^P,$ is shown as follows:

$$F_i^P=F_{\delta }\times t_i^s$$

(9)

(2)

Carbon emissions

The actual carbon emissions of a vessel, $E_i,$ is the product of the fuel consumption and the carbon emission factor ${\eta }_{co2}$:

$$E_i=(F_i^M+F_i^A+F_i^P)\times {\eta }_{co2}$$

(10)

The total actual carbon emissions, $E,$ consists of two parts: the carbon emissions when the vessel is sailing, and the carbon emissions when the vessel dwells at ports of call, as follows:

$$E=\sum _{i\in \mathcal{B}}{E}_i^s+\sum _{i\in N}{E}_i^p$$

(11)

Similarly, the total EU-related carbon emissions can be obtained as follows:

$$E^{EU}=\sum E_i^{EU-s}+\sum E_i^{EU-p}$$

(12)

$$E_i^{EU-s}=\left\{\begin{array}{c}50\%{\eta }_{co2}\times \left(F_i^M+F_i^A\right), \mathrm{voyage} \mathrm{starting} \mathrm{or} \mathrm{ending} \mathrm{at} \mathrm{EU} \mathrm{ports} \mathrm{on} \mathrm{sengment} i \\ 100\%{\eta }_{co2}\times \left(F_i^M+F_i^A\right), \mathrm{sengment} i \mathrm{is} \mathrm{in} \mathrm{the} \mathrm{EU} \mathrm{region} \end{array}\right.$$

(13)

$$E_i^{EU-p}=100\mathrm{\%}{\eta }_{co2}\times F_i^P,i\in Y$$

(14)

where $E_i^{EU-s}$ is the EU-related carbon emissions from a vessel sailing on segment i of the shipping route, $E_i^{EU-p}$ is the EU-related carbon emissions from a vessel dwelling at EU port i $\mathrm{a}\mathrm{n}\mathrm{d} Y$ is the set of EU ports.

(3)

Carbon emission cost, C_A

The carbon emission cost paid by liner companies depends on the total EU-related carbon emissions $E^{EU}$, the carbon price $\varphi$ and the proportion of carbon emissions included in the EU environmental policy $\lambda$, as follows:

$$C_A=E^{EU}\varphi \lambda =(\sum E_i^{EU-s}+\sum E_i^{EU-p})\times \varphi \times \lambda$$

(15)

4.

Bunker fuel cost, C_F

The study examines a vessel refueling at ports of call, where bunker fuel cost depends on fuel volume and fuel price at each port. These costs include the amount of MGO ($w_i^{MGO}$) and VLSFO ($w_i^{VLSFO}$) refueled at port i, along with the fuel prices ($C_i^{MGO}$ and $C_i^{VLSFO}$).

The bunkering amounts are calculated from the fuel stock when a vessel arrives at port $i$ ($y_i^{1MGO}$ or $y_i^{1VLSFO}$) and departs from port $i$ ($y_i^{2MGO}$ or $y_i^{2VLSFO}$), and the amount of fuel consumed by auxiliary engines, as follows:

$$w_i^{MGO}=y_i^{2MGO}-y_i^{1MGO}+F_{\delta }\times t_i^s$$

(16)

$$w_i^{VLSFO}=y_i^{2VLSFO}-y_i^{1VLSFO}$$

(17)

Thus, the bunker fuel cost $C_F$ for a vessel on a voyage is as follows:

$$C_F=\sum _{i=1}^n{(x}_i^{MGO}{C}_i^{MGO}{w}_i^{MGO}+x_i^{VLSFO}{C}_i^{VLSFO}{w}_i^{VLSFO}+C_g{(x}_i^{MGO}+x_i^{VLSFO}\left)\right)$$

(18)

2.2.3. Model Formulation

Based on the formulas for each cost in the previous section, $\mathrm{M}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l} 1$ is formulated to minimize the operating cost of the liner service.

$$\begin{array}{ll}[\mathrm{M}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l} 1]: min& (C_{ZE}+C_T+C_A+C_F)\\ & =k{C}_E+{\displaystyle \sum _{i=1}^n}\psi {\left[t_i^a-T_i^l\right]}^{+}+(\sum E_i^{EU-s}+\sum E_i^{EU-p})\times \varphi \times \lambda \\ & +{\displaystyle \sum _{i=1}^n}{(x}_i^{MGO}{C}_i^{MGO}{w}_i^{MGO}+x_i^{VLSFO}{C}_i^{VLSFO}{w}_i^{VLSFO}+C_g{(x}_i^{MGO}+x_i^{VLSFO}\left)\right)\end{array}$$

(19)

subject to

$$\sum _{i=1}^n({\displaystyle \frac{S_i^E}{V_i^E}}+{\displaystyle \frac{S_i^N}{V_i^N}}+t_i^s)\le 168k,$$

(20)

$$w_i^{MGO}=y_i^{2MGO}-y_i^{1MGO}+F_{\delta }\times t_i^s,\forall i\in N$$

(21)

$$w_i^{VLSFO}=y_i^{2VLSFO}-y_i^{1VLSFO},\forall i\in N$$

(22)

$${20\mathrm{\%}{x}_i^{MGO}{H}^{MGO}\le w}_i^{MGO}\le x_i^{MGO}{H}^{MGO},\forall i\in N$$

(23)

$${20\mathrm{\%}{x}_i^{VLSFO}{H}^{VLSFO}\le w}_i^{VLSFO}\le x_i^{VLSFO}{H}^{VLSFO},\forall i\in N$$

(24)

$$y_i^{1MGO}\ge 10\mathrm{\%}{H}^{MGO},\forall i\in N$$

(25)

$$y_i^{1VLSFO}\ge 10\mathrm{\%}{H}^{VLSFO},\forall i\in N$$

(26)

$$y_i^{1MGO}+w_i^{MGO}\le H^{MGO},\forall i\in N$$

(27)

$$y_i^{1VLSFO}+w_i^{\mathrm{V}\mathrm{L}\mathrm{S}\mathrm{F}\mathrm{O}}\le H^{VLSFO},\forall i\in N$$

(28)

$$y_1^{1MGO}=1000,$$

(29)

$$y_1^{1VLSFO}=1000,$$

(30)

$$y_{i+1}^{1MGO}=y_i^{2MGO}-M_i\left(V_i^E\right)S_i^E-F_{\delta }\times \left({\displaystyle \frac{S_i^E}{V_i^E}}+{\displaystyle \frac{S_i^N}{V_i^N}}\right),\forall i<n$$

(31)

$$y_{i+1}^{1VLSFO}=y_i^{2VLSFO}-M_i\left(V_i^N\right)S_i^N,\forall i<n$$

(32)

$$y_1^{1MGO}=y_i^{2MGO}-M_i\left(V_i^E\right)S_i^E-F_{\delta }\times ({\displaystyle \frac{S_i^E}{V_i^E}}+{\displaystyle \frac{S_i^N}{V_i^N}}),i=n$$

(33)

$$y_1^{1VLSFO}=y_i^{2VLSFO}-M_i\left(V_i^N\right)S_i^N,i=n$$

(34)

$$t_1^a=0$$

(35)

$$t_{i+1}^a=t_i^a+t_i^s+{\displaystyle \frac{S_i^E}{V_i^E}}+{\displaystyle \frac{S_i^N}{V_i^N}},\forall i<n$$

(36)

$$t_{n+1}^a=T$$

(37)

$$k\le k_{max}$$

(38)

$$\sum _{i=1}^n(x_i^{MGO}+x_i^{VLSFO})\le q_{max}$$

(39)

$$V_{min}\le V_i^E\le V_i^N\le V_{max},\forall i\in N$$

(40)

$$k\in Z^{+}$$

(41)

$$x_i^{MGO},x_i^{VLSFO}\in \left\{\mathrm{0,1}\right\},\forall i\in N$$

(42)

The objective function (19) is derived to minimize the operating cost. Constraint (20) ensures weekly service frequency. Constraints (21)–(24) address the amounts of MGO and VLSFO to be refueled at port i and the restrictions on bunkering amounts. Constraints (25) and (26) set the safe stock upon arrival. Constraints (27) and (28) denote the maximum capacity for the bunker tanks. Constraints (29) and (30) specify fuel stocks at the port of origin. Constraints (31)–(34) indicate the remaining fuel upon arrival. Constraints (35)–(37) represent arrival time constraints. Constraint (38) restricts the number of allocated ships. Constraint (39) refers to the maximum number of refueling operations on a voyage. Constraint (40) establishes speed limits inside and outside ECAs. Constraint (41) indicates that the number of vessels required to be assigned is a positive integer. Constraint (42) lists decision variables, indicating whether or not to refuel at the port.

2.3. Model Transformation

Since the fuel consumption cost function is convex [15,33,34], a segmented linear approximation is performed. The steps are as follows:

Step 1. For the non-smooth term ${[t_i^a-T_i^l]}^{+}$, introducing the auxiliary variable ${\beta }_i$, such that ${\beta }_i={[t_i^a-T_i^l]}^{+}$, thus

$$min\sum _{i=1}^n{[t_i^a-T_i^l]}^{+}=min\sum _{i=1}^n\mathrm{m}\mathrm{a}\mathrm{x}\{t_i^a-T_i^l,0\}=min\sum _{i=1}^n{\beta }_i$$

(43)

subject to

$${\beta }_i\ge t_i^a-T_i^l$$

(44)

$${\beta }_i\ge 0$$

(45)

Step 2. Introducing the variable $u_i$ to Equations (4) and (5) such that $u_i^E={\displaystyle \frac{1}{V_i^E}}$, $u_i^N={\displaystyle \frac{1}{V_i^N}}$. To simplify the fuel consumption functions, let the constant terms in the fuel consumption functions be replaced by $R=[\alpha +(1-\alpha ){\displaystyle \frac{\omega }{{\omega }_0}}]F_{\Omega }$. The fuel consumption functions of the vessel sailing on voyage segment $i$ at unit distance are simplified as $M_i(u_i^E)=R\times {\displaystyle \frac{{(u_i^E)}^{-2}}{{{(V}_d)}^3}}$, $M_i(u_i^N)=R\times {\displaystyle \frac{{(u_i^N)}^{-2}}{{{(V}_d)}^3}}$.

Step 3. Given an error $\epsilon$, let $u_i^E\in$ [$u^{min}, u^{max}$], where $u^{min}={\displaystyle \frac{1}{{\mathrm{V}}_{max}}}$, $u^{max}={\displaystyle \frac{1}{{\mathrm{V}}_{min}}}$, $M_i\left(u_i^E\right)\in [R{\displaystyle \frac{({u^{max})}^{-2}}{{{(V}_d)}^3}},R{\displaystyle \frac{(u^{min}{)}^{-2}}{{{(V}_d)}^3}}]$. Within this range, the fuel consumption function is sequentially segmented and linearly approximated. Let $\mathrm{S}=\left\{1, 2, \dots ,m\right\}$ be the set of line segments of the linear approximation of the fuel consumption function. When the approximation error is not greater than ε, $M_i(u_i^E)$ is replaced by ${\overline{M}}_i^r\left(u_i^E\right)$, ${\overline{M}}_i^r\left(u_i^E\right)=M_i\left(u_i^{E,r}\right)+{SL}_r\times \left(u_i^E-u_i^{E,r}\right)$, ${SL}_r={\displaystyle \frac{M_i\left(u_i^{E,r+1}\right)-M_i\left(u_i^{E,r}\right)}{u_i^{E,r+1}-u_i^{E,r}}}$, $\forall i\in N,r\in \mathrm{S}$.

Based on the “Big M” method [19], Model 1 can be reformulated as

$$\begin{array}{cc}\left[\mathrm{M}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}2\right]:& \mathit{min}\left(C_{ZE}+C_T+C_A+C_F\right)\\ & =k{C}_E+{\displaystyle \sum _{i=1}^n}\psi \times {\beta }_i+(\sum E_i^{EU-s}+\sum E_i^{EU-p})\times \varphi \times \lambda \\ & +{\displaystyle \sum _{i=1}^n}{(x}_i^{MGO}{C}_i^{MGO}{w}_i^{MGO}+x_i^{VLSFO}{C}_i^{VLSFO}{w}_i^{VLSFO}+C_g{(x}_i^{MGO}+x_i^{VLSFO}))\end{array}$$

(46)

subject to

$$\sum _{i=1}^n(S_i^E{u}_i^E+S_i^N{u}_i^N+t_i^s)\le 168k,$$

(47)

$$y_{i+1}^{1MGO}=y_i^{2MGO}-{\overline{M}}_i^r\left(u_i^E\right)S_i^E-F_{\delta }\times (S_i^E{u}_i^E+S_i^N{u}_i^N),\forall i<n, r\in S$$

(48)

$$y_{i+1}^{1VLSFO}=y_i^{2VLSFO}-{\overline{M}}_i^r\left(u_i^N\right)S_i^N,\forall i<n, r\in S$$

(49)

$$y_1^{1MGO}=y_i^{2MGO}-{\overline{M}}_i^r\left(u_i^E\right)S_i^E-F_{\delta }\times \left(S_i^E{u}_i^E+S_i^N{u}_i^N\right),i=n,r\in S$$

(50)

$$y_1^{1VLSFO}=y_i^{2VLSFO}-{\overline{M}}_i^r\left(u_i^N\right)S_i^N,i=n,r\in S$$

(51)

$$t_{i+1}^a=t_i^a+t_i^s+S_i^E{u}_i^E+S_i^N{u}_i^N,\forall i<n$$

(52)

$${\displaystyle \frac{1}{V_{max}}}\le u_i^{\mathrm{N}}\le u_i^E\le {\displaystyle \frac{1}{V_{min}}},\forall i\in N$$

(53)

$${\sum }_{r\in S}{b}_i^r=1,\forall i\in N$$

(54)

$$s_r{b}_i^r\le u_i^E,\forall i\in N,r\in S$$

(55)

$$e_r+M^{\prime }\left(1-b_i^r\right)\ge u_i^E,\forall i\in N,r\in S$$

(56)

$${\overline{M}}_i^r\left(u_i^E\right)\ge {SL}_r{u}_i^E+{IN}_r-M^{″}(1-b_i^r),\forall i\in N,r\in S$$

(57)

$$s_r{b}_i^r\le u_i^N,\forall i\in N,r\in S$$

(58)

$$e_r+M^{\prime }(1-b_i^r)\ge u_i^N,\forall i\in N,r\in S$$

(59)

$${\overline{M}}_i^r\left(u_i^N\right)\ge {SL}_r{u}_i^N+{IN}_r-M^{″}\left(1-b_i^r\right),\forall i\in N,r\in S$$

(60)

$$b_i^r\in \left\{\mathrm{0,1}\right\},\forall i\in N,r\in S$$

(61)

Model 2 is subject to the constraints, including Equations (21)–(30), (35), (37)–(39), (41), (42), (44) and (45). The constraints (20), (31)–(34), (36) and (40) in Model 1 are linearized into the new constraints (47)–(53), respectively. Constraints (54)–(61) are new constraints brought about by the use of the “Big M” method. Constraint (54) ensures that only one linear line segment $r$ is chosen to approximate the fuel consumption function for the $i^{th}$ segment. In constraints (55), (56), (58) and (59), $s_r$, $e_r$ represent the horizontal values of the start and end points corresponding to line segment r. Constraints (57) and (60) are used to calculate an approximation of fuel consumption ${\overline{M}}_i^r\left(u_i^E\right)$ or ${\overline{M}}_i^r\left(u_i^N\right)$ when a vessel sailing at speed $u_i^E$ or $u_i^N$ on segment i. ${SL}_r and {IN}_r$ denote the slope and intercept of the linear approximation segment r for fuel consumption, ${SL}_r={\displaystyle \frac{M_i\left(u_i^{E,r+1}\right)-M_i\left(u_i^{E,r}\right)}{u_i^{E,r+1}-u_i^{E,r}}}$, ${IN}_r=M_i\left(u_i^{E,r}\right)-{SL}_r\times u_i^{E,r}$. The positive number $M^{\prime }$ is introduced to ensure the linear segment $r$ approximates the nonlinear function only in a specific range and takes the value of $M^{\prime }=u^{max}.$ $M^{″}$ is used to estimate the approximate fuel consumption and takes the value of $M^{″}={SL}_1{u}^{min}+{IN}_1.$ Constraint (61) indicates that $b_i^r$ is a 0–1 variable, while $b_i^r=1$ indicates that the fuel consumption function $M_i\left(u_i^E\right)$ or $M_i\left(u_i^N\right)$ on segment i can be represented by ${\overline{M}}_i^r\left(u_i^E\right)$ or ${\overline{M}}_i^r\left(u_i^N\right)$ at the linear segment $r$, $r\mathsf{ϵ}S$. Otherwise, $b_i^r=0$. Obviously, Model 2 has been converted to a mixed-intege linear optimization model and can be solved by CPLEX 22.1.1 software [17,18]. The flow chart of the linearization process and solution procedure is shown in Figure 2.

3. Results

This section will conduct a case study on CMA CGM Line’s Asia–Europe liner service, FAL3, which passes through Qingdao, Shanghai, Ningbo and Yantian in sequence and reaches Singapore via the South China Sea to the south; then, it crosses the Indian Ocean and the Suez Canal with stops at Algeciras and arrives at Rotterdam, where a return voyage is initiated to Southampton via Southampton, Antwerp and Le Havre; and finally, returns to Algeciras, Singapore and Qingdao to complete a voyage [35]. Based on the route data of the container liner vessel CMA CGM Antoine de Saint-Exupéry, assigned by CMA CGM [36] and HiFleet Co., Ltd. [37], the details of the liner service are shown in Figure 3.

According to the Clarkson Shipping Intelligence Network, the loading capacity of the container liner CMA CGM Antoine de Saint-Exupéry is 20,600 TEUs (Twenty-foot Equivalent Units). The values of the parameters are shown in

Table 1. Parameters related to the liner shipping service.

Parameter	Value	Ref.
$\mathrm{T}$	2120	[37]
$V_{max}$	24	[37]
${\eta }_{co2}$	3.206	[25,26,38]
${\mathsf{\omega }}_0$	20,600	[39]
$R_M$	206	[27]
$L_M$	0.8	[40]
$P_M$	70,950	[39]
$R_A$	221	[27]
$L_A$	0.5	[40]
$P_A$	13,500	[39]
$\mathsf{\alpha }$	0.92	[41]
$\mathsf{\varphi }$	96.3	[42]

and

Table 2. The values of other parameters.

Parameter	Value	Parameter	Value
$\mathsf{\omega }$	16,480	$V_d$	20
${\mathrm{k}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$	15	$V_{min}$	10
$q_{max}$	6	$C_g$	1000
$\psi$	1000	$H^{MGO}$	3000
$C_E$	180,000	$H^{VLSFO}$	3000
$\epsilon$	0.01	$\lambda$	70%

.

Currently, the Baltic Sea, North America, the North Sea and the U.S. Caribbean Sea belong to the IMO-designated 0.1% m/m ECAs, and the Hainan area, as well as China’s inland waterways, also belongs to the 0.1% m/m ECAs [7]. On 1 May 2025, the Mediterranean Sea will become the fifth ECA with 0.1% m/m [43]. Obviously, global maritime agencies are becoming increasingly strict in controlling the emission of sulfur oxides from ships. Therefore, this study assumes that in the future, the sulfur content of fuel oils used by ships in Mediterranean ECA and China coastal ECA (within 12 nautical miles of the coastal baseline) will be limited to 0.1% m/m and that the port of Southampton in the United Kingdom, which is not located in the EU area, will also need to follow the 0.1% m/m ECA policy. The data for the FAL3 route were recorded in

Table 3. Data on fuel prices and the expanded ECAs that will be implemented in the future.

No.	Port	Distance, N miles		Dwelling Time, Hours	Time Window, Hours		Fuel Price, USD
		$\mathbf{E}\mathbf{C}\mathbf{A}$	$\mathbf{N}\mathbf{o}\mathbf{n}–\mathbf{E}\mathbf{C}\mathbf{A}$		${\mathit{T}}_{\mathit{i}}^{\mathit{e}}$	${\mathit{T}}_{\mathit{i}}^{\mathit{l}}$	MGO	VLSFO
1	Qingdao	138	308	34	0	0	852	653
2	Shanghai	160	0	37	98	100	855	637
3	Ningbo	454	247	37	142	148	817	623
4	Yantian	44	1426	18	252	257	947	645
5	Singapore	0	5022	20	362	368	797	646
6	Suez	1939	88	12	652	660	1090	737
7	Algeciras	407	996	25	786	800	870	635
8	Rotterdam	251	0	119	952	961	819	581
9	Southampton	296	0	80	1095	1099	792	605
10	Antwerp	297	0	34	1210	1217	787	612
11	Le Havre	222	1031	43	1265	1279	810	618
12	Algeciras	1932	0	15	1371	1404	870	635
13	Port Said	0	5107	15	1516	1522	886	648
14	Singapore	33	2436	42	1888	1913	818	646
15	Qingdao	--	--	34	2120	2120	852	653

by collecting the voyage data and measuring it with the mapping tool provided by HiFleet.

The fuel consumption function is linearized, and the results are shown in

Table 4. Sailing speed and bunkering amount choices in the expanded ECAs that will be implemented in the future.

No.	Port	Speed, Knots		Arrival Time, Hours	Whether or Not to Refuel		Bunkering Amount, Tons
		ECA	$\mathbf{N}\mathbf{o}\mathbf{n}–\mathbf{E}\mathbf{C}\mathbf{A}$		MGO	$\mathbf{V}\mathbf{L}\mathbf{S}\mathbf{F}\mathbf{O}$	MGO	$\mathbf{V}\mathbf{L}\mathbf{S}\mathbf{F}\mathbf{O}$
1	Qingdao	14.12	15.38	0	0	0	0	0
2	Shanghai	14.12	--	63.79	0	0	0	0
3	Ningbo	14.12	15.38	112.12	0	1	0	1999.50
4	Yantian	14.12	15.38	197.32	0	0	0	0
5	Singapore	--	15.38	311.12	1	0	2078.65	0
6	Suez	14.12	15.38	657.55	0	0	0	0
7	Algeciras	13.04	14.12	801.82	0	0	0	0
8	Rotterdam	13.04	--	930.06	0	1	0	2637.72
9	Southampton	13.04	--	1068.32	0	0	0	0
10	Antwerp	12.18	--	1171.02	1	0	2700	0
11	Le Havre	13.04	14.12	1229.52	0	0	0	0
12	Algeciras	13.04	--	1362.54	0	0	0	0
13	Port Said	--	14.45	1526.62	0	0	0	0
14	Singapore	13.04	14.12	1903.19	0	1	0	600
15	Qingdao	--	--	2120	--	--	--	--

. Table 4 shows that the optimal speeds of the liner vessel sailing within ECAs are as follows: from east to west, the optimal speed from Qingdao to Yantian is 14.12 knots, from Suez to Rotterdam via Algeciras the optimal speeds are 14.12, 13.04 and 13.04 knots. The optimal speeds along the way back to Antwerp, Le Havre and finally to Algeciras via Southampton are 13.04, 12.18, 13.04 and 13.04 knots, respectively. The optimal speeds for a vessel sailing outside of the ECAs can likewise be calculated. The optimal bunkering strategy is 2078.65 and 2700 tons MGO at Singapore and Antwerp ports, respectively, and 1999.50, 2637.72 and 600 tons VLSFO at Ningbo, Rotterdam and Singapore ports, respectively. The number of container liners allocated to the route is 13. The bunkering ports selected, bunkering amounts determined, and sailing speeds calculated based on the model provide a reference for decision-makers in liner shipping companies.

Figure 4 shows the proportion of each cost in the operating cost: bunker fuel cost accounts for 70.81%, total weekly operational cost for 23.83%, carbon emission cost for 5.10% and time delay penalties for 0.26%. Obviously, bunker fuel cost accounts for the highest percentage. Thus, under increasingly strict emission regulations, shipping companies must optimize sailing speeds and bunkering strategies to maintain cost advantages.

4. Discussion

4.1. Effect of Fuel Tank Capacity

Assume that the fuel tank capacity of MGO and VLSFO are equal, keep other parameters fixed and uniformly adjust the size of the two fuel tanks. Figure 5 and Figure 6 show that changes in fuel tank capacity have a significant effect on the operating cost, actual carbon emissions, total bunkering amounts of MGO and VLSFO.

In Figure 5, the operating cost decreases significantly when the bunker tank capacity changes from 2000 to 3000 tons because the ocean liner has a need to travel a long distance and cannot store enough fuel if the fuel tanks are small, and it will increase the frequency of bunkering to a certain extent. Figure 5 shows that 4000 tons is an important threshold for the liner vessel when making tank size design decisions.

Figure 6 shows that the total bunkering amount of VLSFO is consistently higher than that of MGO. When the fuel tank capacity is increased from 2000 to 3000 tons, the total bunkering amount of MGO decreases substantially, as the increase in tank capacity reduces the number of bunkering operations, resulting in a decrease in the total bunkering amount of MGO on long voyages. Although the total bunkering amount of VLSFO increases, the operating cost does not rise significantly due to its much lower price compared to MGO. When the fuel tank capacity increases to 4000 tons, the total bunkering amount of MGO increases, while the total bunkering amount of VLSFO decreases. When the fuel tank capacity reaches 4000 tons or even larger, both the total bunkering amounts of MGO and VLSFO will remain stable.

From

Table 5. Bunker fuel cost under different fuel tank capacities.

Bunker Tank Capacity, Ton	Bunker Fuel Cost, USD	Bunker Tank Capacity, Ton	Bunker Fuel Cost, USD
2000	7.25 × 106	5000	6.94 × 106
3000	6.95 × 106	6000	6.96 × 106
4000	6.94 × 106	7000	6.97 × 106

, it can be seen that when the bunker tank capacity increases from 2000 tons to 4000 tons, the bunker fuel cost decreases from 7.25 × 10⁶ USD to 6.94 × 10⁶ USD. When the capacity of the bunker tank changes from 5000 tons to 7000 tons, the bunker fuel cost continues to increase.

Based on the results in Figure 5 and Table 5, it can be concluded that the optimal bunker tank capacity for the liner vessel on the route is 4000 tons. Meanwhile, the bunker tank capacity of 5000 tons is the sub-optimal option on this route. Specifically, there exists a bunker tank capacity threshold below which the operating cost will increase significantly. However, once this threshold is reached, the optimal solution can be chosen to realize operating cost savings.

It is estimated that for one voyage, i.e., 2120 h, when the fuel tank capacity is changed from 2000 to 4000 tons, the savings in the operating cost for the liner vessel on the route is 0.34 × 10⁶ USD. When the tank capacity is changed from 3000 to 4000 tons, the savings in operating cost is 0.20 × 10⁵ USD. The Asia–Europe shipping route is one of the busiest and most important trade routes in the world. In particular, FAL3, as one of the important routes, requires dozens of ships to operate cyclically every year. Therefore, from the perspective of shipping companies, it is highly cost-effective to determine an optimal bunker tank capacity that minimizes operating cost.

4.2. Effect of Fuel Price Spread

Fuel prices are constantly changing, and they fluctuate greatly in real time depending on the port. Data from Ship & Bunker show that from 7 April 2021, to 5 April 2024, the global average price of VLSFO ranged from 511.75 USD to 1136.50 USD, while MGO prices ranged from 599.75 USD to 1427.00 USD. In this period, the price spread, ${\delta }_i=C_i^{MGO}-C_i^{VLSFO}$, had a maximum of 400 USD and a minimum of 100 USD, indicating significant price fluctuations. Stricter emission standards have led to an increase in demand for MGO, which will widen the price difference between the two fuel oils. While keeping other parameters and VLSFO prices fixed, four different fuel price spreads {100, 200, 300, 400} were used to perform a sensitivity analysis. The results are shown in Figure 7, Figure 8 and Figure 9.

Figure 7 shows that as the price spread between the two fuel oils increases from 100 to 400 USD, the total operating cost and bunker fuel cost increases, and the proportion of bunker fuel cost in total operating cost increases from 69.48% to 73.42%. The result indicates that when the price spread between the two fuel oils is large, shipping companies should pay more attention to the bunkering strategy and need more refined management approaches to reduce the bunker fuel cost and total operating cost.

In Figure 8, as the fuel price spread ${\delta }_i$ between MGO and VLSFO increases from 100 USD to 400 USD, the average planned speed within ECAs shows an overall decreasing trend, the average speed outside of ECAs first increases and then decreases and the total delay time increases. This is because too high a speed will lead to an increase in fuel consumption, which, in turn, will increase bunker fuel cost, and when the price difference is too large, the vessel will have to reduce its speed within ECAs and reduce the bunkering amount of MGO to ensure economic efficiency. Liner operators should dynamically determine sailing speeds, bunkering ports and bunkering amounts according to the fuel price spread and delay penalty cost.

Figure 9 depicts how shipping companies choose refueling ports and bunkering amounts at ports under different price differences. The bunkering strategies adopted by liner operators are different for different fuel price spreads. As the fuel price spread increases from 100 USD to 200 USD, the bunkering strategy of the shipping company has also been adjusted: the liner company will refuel MGO at the port of Antwerp instead of Le Havre. Shipping companies should pay close attention to the dynamic changes in fuel prices at bunkering ports and adjust their bunkering strategies in a timely manner, according to the differences in fuel prices, to obtain economic benefits and comply with ship emission regulations.

5. Conclusions

This study proposes an optimization approach to minimize the operating cost of an ocean liner while complying with ship emission regulations. Quantifying the operating cost, including the total weekly operational cost, time delay penalty cost, carbon emission cost and bunker fuel cost, gives relevant departments a clear understanding of a liner company’s expenses. A mixed-integer nonlinear optimization model is then developed, and the nonlinear components are linearized. The research takes the Asia–Europe route FAL3 as a case study and combined with ECA policies, obtains the planned sailing speed and refueling strategies. Finally, this study examines how fuel tank capacity and fuel price spread affect various costs, CO₂ emissions, planned speeds and refueling strategies. The model proposed in this study plays an important role in optimizing the fuel tank capacity, as well as the bunkering strategy, under different fuel price spreads.

The decision-making suggestions for liner operators are provided and summarized as follows:

Although increasing the capacity of fuel oil tanks can reduce the number of refueling operations and minimize the operating cost. However, a bunker tank capacity that is too large will affect the cargo space and reduce economic benefits. Based on the actual loading capacity, with the help of the proposed model, ship operators can obtain the optimal oil tank capacity to reduce costs while complying with ship emission regulations.

Dynamic changes in fuel oil prices will lead to changes in fuel costs, planned sailing speeds, and carbon dioxide emissions. Liner operators could use the model proposed in this study, combined with predicted fuel oil prices and real load capacity, to adjust refueling strategies and planned sailing speeds in a timely manner in order to balance service quality and costs and avoid simply pursuing the lowest operating cost that may affect shipping punctuality.

The proposed method is widely applicable, allowing shipping companies to use it once the vessel and route data are available, thereby ensuring the model’s practicality. Real data from the liner vessel CMA CGM Antoine de Saint-Exupéry confirms the feasibility of the proposal. Shipping companies could adjust the model to different environmental regulations, routes and vessels to enhance decision support. By optimizing sailing speeds and bunkering strategies, the model can ensure compliance with environmental standards while achieving economic objectives. The values of parameters in the model can be adjusted according to changes in actual ECA policies. For instance, if the scenario occurs in 2026 or later, setting the parameter $\lambda$ to 100% will enable the model to adapt to a wider range of scenarios.

The model also has some limitations: although the value of the carbon emission factor was taken from the literature [41], and the data on the distance between ports were provided by HiFleet [37], these values are still slightly different from the actual situation, which has a weak impact on the calculation of the operating cost.

Future research involves the following two aspects: Considering the impact of weather conditions on the planned speeds and bunkering strategies will make the research more realistic. On the other hand, since the price of fuel oil is affected by many factors and fluctuates greatly, the accurate prediction of fuel oil price and obtaining the optimal bunkering strategy should receive extensive attention.