Optimization of Fuel Consumption for an Offshore Supply Tug Using a Backtracking Algorithm

Abstract

This paper introduces a backtracking algorithm for the fuel optimization of an offshore supply tugboat. The aim is to determine the optimal cruising speed that minimizes fuel use under operational constraints. Many older vessels in local offshore regions face limitations in adopting new fuel efficient technologies due to financial constraints. Hence, alternative cost-effective methods are needed to improve energy use and reduce emissions from these older ships. We propose using a backtracking algorithm to systematically explore all potential speed solutions and find the optimal one. Operational constraints like time restrictions and weather factors are incorporated during the optimization. The algorithm branches out to potential solutions and backtracks when they violate constraints. This allows for the pruning of infeasible solutions to improve the computational efficiency. The study provides the basis for optimizing offshore voyages as a sustainable transportation activity. Further work could expand the technique by adding parameters and real-time data.

1. Introduction

According to the Fourth GHG Study 2020 [1], released by the International Maritime Organization (IMO), the shipping industry accounted for 2.89% of global CO₂ emissions in 2018, emitting a total of 1056 million tons. International shipping alone contributed 740 million tons of CO₂ in 2018. The study predicts a potential increase in emissions of 90–130% by 2050 based on long-term economic and energy scenarios, necessitating urgent improvements in energy efficiency and emission reductions. To address this challenge, the IMO is actively working towards improving energy efficiency and reducing greenhouse gas (GHG) emissions through a global approach [2]. Their efforts include developing emission reduction measures, providing technical support, and fostering capacity-building activities. While new technologies are being adopted for newer ships, many older offshore vessels face financial limitations that hinder tech upgrades. Hence, cost-effective optimization methods like voyage planning, weather routing, and speed optimization are vital for these ships.

The objective of reducing the carbon intensity of international shipping is to decrease CO₂ emissions per unit of transport work. This involves enhancing ship energy efficiency and reducing the carbon footprint of the shipping industry. The term “transport work” refers to the function of ships and vessels primarily involved in commercial transportation of passengers and/or cargo between ports. However, it excludes “working/service vessels” used in offshore and marine contracting, which primarily support offshore industries and marine construction.

For the offshore sector, the International Marine Contractors Association (IMCA) has proposed two proxies to measure carbon intensity. Proxy A considers yearly energy consumption, installed power, running hours, and CO₂ emissions derived from fuel consumption, while Proxy B focuses on vessel operational utilization time and corresponding CO₂ emissions. It is important to note that both proxies rely on accurate and reliable data regarding energy consumption, CO₂ emissions, engine ratings, running hours, and operational utilization time [3]. The aim is to develop energy efficiency measures for working vessels, given that existing performance systems are intended for long-distance voyages and do not fit well to these ships [4].

The global market for offshore supply vessels is experiencing significant growth due to increasing demand for sustainable energy and policies supporting low-carbon power technologies. Fuel optimization is particularly advantageous for the supply vessels [5].

Prior works have applied optimization algorithms like dynamic programming and calculus of variations for voyage optimization. However, these mathematical programming methods have limitations in handling complex operational constraints. Backtracking provides a flexible discrete optimization approach to systematically explore all solutions while accommodating constraints. Despite its potential, backtracking has been underutilized in maritime applications compared to other industries.

In this study, we demonstrate the advantages of using a backtracking algorithm for voyage optimization under operational constraints. The algorithm efficiently prunes infeasible solutions that violate constraints, thereby enhancing the computational performance. This allows for the comprehensive optimization of voyages for minimal fuel consumption unlike traditional methods. The incorporation of detailed parameters, like weather data and vessel loading, also improves the optimization accuracy.

This paper’s key contributions include the introduction of backtracking for maritime voyage speed optimization and implementation under variational constraints. The study provides a basis for the further development of backtracking techniques to support sustainable offshore transportation. Overall, the paper highlights the promising potential of discrete optimization algorithms like backtracking for critical maritime applications.

2. Literature Review

To minimize fuel consumption and CO₂ emissions, as well as to achieve high energy efficiency levels, the maritime industry must explore various voyage optimization methods.

Solutions such as voyage planning and speed optimization taking into account weather routing have been studied, and the results demonstrate an overall improvement in energy efficiency reducing fuel consumption and emissions. Weather routing algorithms with the objective of minimizing fuel consumption have been proposed. Optimization algorithms such as dynamic programming were firstly used in a two-dimensional approach by de Wit [6] and Calvert et al. [7]. A step forward was implemented by Shao, Zhou, and Thong (2012) [8], who developed a 3D dynamic programming method considering the ship power and heading towards the minimization of fuel consumption. Both approaches are based on Bellman’s principle of optimality [9], on the idea that the course set-up is optimal if the previous solution was optimal at each stage.

Calculus of variations, discrete optimization methods based on grid approaches, evolutionary algorithms, and other modeling and ship weather approaches have been used to solve the challenge of choosing an optimal voyage path [10].

While these continuous optimization methods have shown some success, they have limitations in handling real-world operational constraints and restrictions. For instance, dynamic programming suffers from the “curse of dimensionality” in high-dimensional solution spaces [11].

Through the analysis of several fuel consumption optimization algorithms, Psaraftis demonstrated that by taking into account the sailing environment, fuel prices, and other pertinent factors, optimizing the speed of a ship can yield significant energy conservation and emission reduction benefits. A commonly employed approximation of the correlation between fuel consumption and vessel speed is a cubic function per unit of time [12].

Li et al. introduced a speed optimization model designed for a predetermined ship route, aiming to minimize the overall fuel consumption throughout the entire voyage. Their model incorporates the impact of ocean currents and further develops a minimum fuel consumption model. This model can optimize the vessel speed while considering irregular environmental forces. The research utilized mathematical and simulation models, specifically nonlinear speed optimization, as outlined in their work [13].

The progress in computer science and artificial intelligence has resulted in an increased adoption of machine learning techniques for constructing models aimed at improving ship energy efficiency. Lin et al. conducted a study that emphasized the potential benefits of integrating machine learning with optimization algorithms to enhance ship performance and energy efficiency [14]. On a related note, Tarelko et al. focused on the utilization of artificial neural networks (ANNs) to analyze speed and fuel models [15].

In the offshore sector, there is currently an estimated total of 14,000 ships, accounting for approximately 22% of the global fleet with a gross tonnage (GRT) below 400. Most of these ships are classified as tugs and coastal vessels [16].

In the field of shorter distance voyages, research indicates that emphasizing enhancements in propulsion efficiency and exploring alternative methods, such as constructing new vessels with battery-powered electric propulsion, can be advantageous. Liu et al., presented a hybrid electric ship energy efficiency optimization model that takes into account time-varying environmental factors. This model optimizes the real-time energy index considering wind and wave conditions, while also adhering to speed limitations. The findings demonstrate a reduction in fuel consumption by 13.4% and a decrease in the real-time index by 15.2% [17]. The transition to the architecture of an integrated ship system with a DC main bus allows, due to the fuel savings, for an increase in the efficiency of the ship as a whole by 20%, reducing the weight and volume of on-board electrical equipment by 30% [18].

In his article, Jae-Gon et al. studied the ship speed optimization problem with the objective of minimizing the total fuel consumption and derived the intrinsic properties of the problem and developed an exact algorithm based on the properties using a nonlinear mixed integer program [19]. Somakumar et al. proposed a metaheuristic algorithm to optimize both the economic and emission costs for a microgrid that comprised a diesel engine, various other sources, and loads [20].

Windover et al. conducted an analysis of a short sea shipping model and indicated that the fuel savings are highly dependent on the application [21]. Discrete optimization techniques like backtracking provide an alternative approach better suited to complex combinatorial problems with variables that take on discrete values. However, applications in the maritime literature remain relatively scarce. In the maritime field, the backtracking algorithm, informed by a constraint-based approach, was used by Pasaribu et al. [22] to comprehensively explore solution trees to address scheduling challenges in ports, influenced by factors such as captain availability and cargo volume.

This study addresses this research gap by implementing backtracking for an offshore supply vessel voyage. It demonstrates how backtracking can efficiently navigate the combinatorial solution space of speed selections under constraints like time limits, weather disruptions, and vessel loading conditions. The backtracking algorithm may be employed to create a voyage plan that incorporates speed set-up conditions for optimal fuel usage, particularly for short, coastal cruises, by utilizing the vessel’s parameters and the speed/fuel consumption curve.

The promising results provide a foundation for further studies exploring backtracking and other discrete methods like metaheuristics.

3. Materials and Methods

A platform supply vessel is used to transport goods and personnel from a port facility to different offshore structures, such as drilling rigs or production platforms. Typically, the voyage is considered a closed loop and provides a circular linear transport with multiple stops. The voyage involves choosing different speeds that vary from full speed to zero in certain situations, where waiting times are requested by the rigs (Figure 1). The ship cruises at full speed during most cruising intervals but at lower speeds due to restricting regulations or when approaching or departing from a port.

Each voyage segment can be split into smaller segments based on the speed restrictions such as entering or leaving a port.

We can see from Figure 2 that a working vessel, such as an offshore tug, has the utmost impact over the environment during cruising or slow transit phases, which accounts for about 65% of the operational phases. The speed optimization of these phases makes sense as having the most impact over the total fuel consumption and CO₂ emissions.

3.1. Algorithm Suitability Analysis

The proposed algorithm is part of the travelling salesman problem (TSP) family, which is one of the most studied and iconic problems in the field of operations research and computer science. Our objective was to determine the optimal speed for a particular voyage, given a set of stops (oil rigs) and the distances between each pair of stops. The difficulty of the issue grows factorially with the number of visits. The number of stops and other journey characteristics will heavily influence the algorithms used.

Using the backtracking algorithm looks to be a reasonable option when thinking about the specifics of our application, which prioritizes quick coastal voyages with few stops within a limited time constraint.

At its essence, backtracking is an exhaustive search for the optimal solution configuration. The solution space for fuel optimization includes a wide variety of possible inter-segment speeds. By going backwards, you may test different permutations of speed and see how they affect fuel consumption. Backtracking seeks to identify the precise best solution, as opposed to heuristic approaches which yield approximations. This guarantees the optimum time- and energy-saving route and speed set-up between segments.

However, it is important to note that as the number of segments increases, the solution space grows exponentially. While backtracking does eliminate many nonoptimal paths, for a large number of segments, the computational time can become prohibitive, and this can limit the applicative domain.

3.2. Mathematical Model

Given a voyage split into $N$ segments, the objective is to minimize the total fuel consumption, $m_f$. For each segment, the fuel consumption is calculated as:

$$m_{f,i}=\underset{t_i}{\overset{t_{i+1}}{\int }}{m}_f\left(t\right)dt$$

(1)

The total fuel consumption for the voyage is then:

$$m_f=\sum _{i=1}^N{m}_{f,i}$$

(2)

Notation list:

• $m_{f,i}$—Fuel consumption for the $i^{th}$ segment;
• $m_f\left(t\right)$—Fuel flow rate as a function of time;
• $t_i$—Voyage segment starting time;
• $t_{i+1}$—Voyage segment end time.

In this study, the optimization of the fuel consumption is taken as the objective function.

The fuel consumption of a vessel can be calculated by multiplying the fuel flow rate by the time for which the fuel is consumed:

$$m_f=m·t$$

(3)

where $m$ is the mass flow rate (kg/h), and $t$ represents the time (h).

This rate is influenced by several factors, including the ship’s speed, the design of the ship and its engines, and the load being carried. An initial estimation of the ship’s speed can be calculated using the admiralty coefficient of power:

$$A_c=\frac{{∆}^{2/3}·v^3}{P}$$

(4)

where:

• $∆$ represents the displacement of the vessel (typically in tons). It provides an indication of the vessel’s size and weight. The term ${∆}^{2/3}$ is used because it scales more appropriately with the wetted surface area of the vessel, which, in turn, influences the resistance and propulsion needs;
• $v$ represents the speed of the vessel (often in knots). The cube of the speed is used because the power needed to propel a ship generally increases with the cube of its speed, especially as it approaches the hull speed (a speed limit for displacement hulls);
• $P$ is the shaft power of the propulsion system (in horsepower or kilowatts). This represents the power delivered to the propeller.

The admiralty coefficient provides an indication of the efficiency of the ship’s design. A lower value of the coefficient typically suggests a more efficient design in terms of the power required for a given speed and displacement. Essentially, shipbuilders use this coefficient to estimate the power needs of a vessel at specific speeds. By understanding these power requirements, one can then correlate it with the engine’s specific fuel consumption. This correlation provides a clearer picture of the anticipated fuel consumption for the vessel at those speeds. In simpler terms, $A_c$ helps to determine how much fuel a ship might burn based on its design and desired operating speed.

3.3. Backtracking Algorithm

The backtrack technique organizes exhaustive searches by trying to extend partial solutions. If extending is not possible, it reverts to a shorter solution and retries. First termed “backtrack” by D.H. Lehmer in the 1950s, this method has been applied to various combinatorial problems like parsing, game playing, and optimization, having been discovered multiple times over the years. The backtracking algorithm performs a systematic search through all possible combinations within a solutions space [23,24]. A partial solution is defined by a combination of variables satisfying a set of constraints. The algorithm incrementally extends a partial solution towards a complete one in which all variables are assigned. All intermediate combinations are tested to complete the problem constraints and optimal allocation. Whenever the test fails, the algorithm backtracks to the most recently feasible partial solution. The algorithm terminates when all variables are successfully assigned.

In the case of the optimization of the shipping voyage, the potential solution is defined by the assignment of speed values for the different voyage stages:

$$v=\left[v_1, v_2, \dots ,v_N\right]$$

(5)

where $v_i\in X_i=\left\{v_{i1},v_{i2}, \dots , v_{i{M}_i}\right\}$ is the speed for stage i (i = 1, …, N) of the voyage and can take values within a predefined set of speeds. The optimum solution minimizes the goal function representing the fuel consumption during the voyage.

A time constraint is added, considering that the voyage duration should not exceed the value $T$:

$$\sum _{i=1}^N{t}_i\left(v_i,{∆v}_i\right)\le T$$

(6)

where $t_i\left(v_i,{∆v}_i\right)$ is the time required to travel the stage i with the propeller speed $v_i$ and the adjustment of the speed ${∆v}_i$ due to the waves’ action and other environmental conditions.

Each time a feasible solution is found, the minimum of the fuel consumption function $\left(\mathrm{C}\right)$ among all found solutions is revisited. This is useful during the branching procedure. Each time a partial solution $\left[v_1, v_2, \dots ,v_k\right]$ (k < N) has a consumption higher than $C$ or a partial voyage duration greater than $T$, we prune the search tree, since all of the subtrees rooted on $\left[v_1, v_2, \dots ,v_k\right]$ are not considered, and the algorithm is tracking back to the previous partial solution. Thus, the solution space and the computing time are reduced.

The recurrent proposed backtracking procedure for the optimization of the ship voyage can be generically described as follows (Algorithm 1):

Algorithm 1: Bactracking for speed optimization

1. Procedure Branch_Backtrack (k, consumption, time) {

2.   if k = N {

3.     $\mathrm{if} (\mathrm{consumption} <=C$) {

4.     $C$ = consumption

5.     keep as a new feasible solution

6.     }

7.   else {

8. for i = 1 to M {

9.     $\mathrm{if} (\mathrm{consumption}+c_{ki}$$<=C$$\mathrm{and} \mathrm{time}+t_{ki}$$<=T$) {

10.       $\mathrm{Branch}\_\mathrm{Backtrack} (k+1, \mathrm{consumption}+c_{ki}$$, \mathrm{time}+t_{ki}$)

11.      }

12.     }

13.    }

14. }

$15. \mathrm{Branch}\_\mathrm{Backtrack} (1,\infty$,0)

where:

• N—The number of voyage stages;
• M—The number of possible speed values;
• k—The current voyage stage (where 1 is the starting point of the voyage, and N is the final destination);
• Consumption—The current partial fuel consumption;
• Time—The current partial voyage time;
• $c_{ki}$—The consumption required to advance from stage k to stage k + 1 (next link in the path) with the propeller speed $v_i$;
• $t_{ki}$—The travel time to advance from stage k to stage k + 1 (next link in the path) with the propeller speed $v_i$ and the ${∆v}_{ik}$ speed adjustment due to waves and environmental conditions during travel from stage k to k + 1.

Algorithm 1 is designed to efficiently optimize the speed with the objective of minimizing the fuel consumption. It operates by navigating various decision points, symbolized by k, to explore all potential speeds up to M. Through recursive exploration, the algorithm assesses all possible paths, but strategically discards those surpassing the defined consumption threshold c or time constraint t. Upon completion, c represents the lowest consumption satisfying the constraints. The algorithm is initiated by calling Branch_Backtrack with starting values of 1 for k, infinity for consumption, and 0 for time.

3.4. Weather Factor

The resistance of a ship in still water, or “hull resistance”, is the force that opposes the motion of a ship as it moves through the water. This resistance arises primarily from three sources:

• Frictional resistance (R_f): This is the resistance caused by the friction between the water and the ship’s hull. It is mostly influenced by the ship’s size, shape, and the type of surface finish. It generally increases with the square of the speed of the ship;
• Wave-making resistance (R_w): As a ship moves through water, it creates waves. The energy required to create these waves results in a resistance force on the ship. The wave-making resistance is very speed dependent and is especially significant at the ship’s “hump speed” (where the ship’s length is approximately equal to the wavelength of the generated waves);
• Form resistance (R_form): This is due to the shape of the ship. Some shapes are more streamlined than others and, therefore, have less resistance.

The total resistance, $R_t$, of a ship in still water can be approximated as the sum of these three resistances:

$$R_t=R_f+R_w+R_{form}$$

(7)

Some of the key ways weather can affect a vessel are wind effects over the superstructure and sea conditions, like waves and currents. Even though there are different methods for calculating the impact of weather on the ship speed losses, the methodology posited by Kwon (2008) stands out [25]. Predicated on its inherent simplicity, the Kwon approach proves particularly apt for assessing the unique dynamics of supply vessels undertaking coastal journeys. Notably, Kwon’s methodology eliminates the need for intricate ship details like hull lines. The simplicity, applicability, versatility, and speed of this method stand out.

Loss of ship speed under different weather conditions (wind and sea) can be calculated as:

$$\frac{∆V}{V_1}100\%=C_{\beta }{C}_U{C}_{form}$$

(8)

$$∆V=V_1-V_2$$

(9)

$$V_1=F_n\sqrt{L_{pp}g}$$

(10)

where:

• $F_n$ = Froude number;
• $∆V$ = Loss of ship speed (m/s);
• $V_1$ = Ship speed in calm water (m/s);
• $V_2$ = Ship speed under the selected weather conditions (m/s);
• $C_{\beta }$ = Speed direction reduction coefficient, which is dependent on the direction of the weather and the Beaufort number BN (

  Table 1. Reduction coefficient $C_{\beta }$ [25].

  Direction	Angle β	$\mathbf{Reduction} \mathbf{Coefficient} {\mathit{C}}_{\mathit{\beta }}$
  Head Sea	0–30°	$C_{\beta }=1$
  Bow Sea	30°–60°	$C_{\beta }=\frac{1.7 - 0.03\left({\left(BN - 4\right)}^2\right)}{2}$
  Beam Sea	60°–150°	$C_{\beta }=\frac{0.9 - 0.06\left({\left(BN - 6\right)}^2\right)}{2}$
  Aft Sea	150°–180°	$C_{\beta }=\frac{0.4 - 0.03\left({\left(BN - 8\right)}^2\right)}{2}$

  );
• $C_U$ = Speed reduction coefficient, which is dependent on ship’s block coefficient CB, the loading conditions and the Froude number Fn (

  Table 2. Speed reduction coefficient $C_U$ [25].

  Block Coefficient CB	Ship Loading Conditions	Speed Reduction Coefficient CU
  0.55	Normal	$1.7-1.4F_n-7.4({F_n)}^2$
  0.60	Normal	$2.2-2.5F_n-9.7({F_n)}^2$
  0.65	Normal	$2.6-3.7F_n-11.6({F_n)}^2$
  0.70	Normal	$3.1-5.3F_n-12.4({F_n)}^2$
  0.75	Full load or normal	$2.4-10.6F_n-9.5({F_n)}^2$
  0.80	Full load or normal	$2.6-13.1F_n-15.1({F_n)}^2$
  0.85	Full load or normal	$3.1-18.7F_n-28.0({F_n)}^2$
  0.75	Ballast	$2.6-12.5F_n-13.5({F_n)}^2$
  0.80	Ballast	$3.0-16.3F_n-21.6({F_n)}^2$
  0.85	Ballast	$3.4-20.9F_n-31.8({F_n)}^2$

  );
• $C_{form}=$ Hull form coefficient, which is dependent on the ship type, the BN, and the ship displacement $∆$ (

  Table 3. Ship form coefficient C_form [25].

  Ship Type and Loading Conditions	Ship Form Coefficient Cform
  All ships (except container ships) in full load conditions	$0.5BN+\frac{{BN}^{6.5}}{2.7{∆}^{2/3}}$
  All ships (except container ships) in ballast conditions	$0.7BN+\frac{{BN}^{6.5}}{2.7{∆}^{2/3}}$
  Container ships in normal loading conditions	$0.5BN+\frac{{BN}^{6.5}}{22{∆}^{2/3}}$

  );
• BN—Beaufort (nondimensional) number (Figure 3).

Depending on the BN scale, the speed and sea conditions along the sailing route vary for each segment of the voyage (Figure 3).

3.5. Practical Implementation

To implement an optimized speed strategy for a PSV operation, a systematic framework is essential. The process starts with pre-voyage planning, where the voyage route is segmented based on ports, stops, and known speed restrictions. Historical and forecasted weather data for the entire route are gathered. Using the backtracking algorithm, the optimal speed for each segment is calculated, factoring in weather conditions and other constraints, as presented in Section 3.4, fuel consumption predictions are made using mathematical models based on the ship consumption curve and the characteristics of energy requirements with the data found in the ship’s powerplant manual.

Once the voyage is underway, real-time monitoring becomes crucial. The onboard systems must be established to monitor the vessel’s real-time speed and fuel consumption. As the voyage progresses, weather updates should be continually tracked, adjusting planned speeds using Kwon’s formula when significant weather changes are detected.

After the voyage, a post-voyage analysis is conducted. Data on the actual speed profile and fuel consumption are gathered and compared with pre-voyage predictions.

To ensure the long-term success of this strategy, a focus on continuous improvement is necessary. After several voyages, accumulated data should be analyzed to discern patterns and trends.

Some major challenges and limitations include accurately segmenting the route and the unpredictability of weather forecasts, real-time voyage monitoring and dynamic weather conditions, ensuring accurate data collection for post-voyage analysis, and balancing the need for optimization with operational and business constraints.

3.6. Numerical Application

In order to showcase the optimization of the speed using the backtrack algorithm, we utilized a specific voyage represented as a waypoint pathway (Figure 4). The voyage starts at Constanța, the designated home port, and follows recommended routes and waypoints to ensure adherence to coastal navigation regulations.

During the voyage, there are two stops: one at the Istria block (R18) to supply the Central platform and another at the ANA platform (R5). The endpoint of the voyage is the Constanța port.

The entire voyage is divided into segments, starting from R11 and concluding at R10, while passing through following waypoints, in this order: R8, R9, R12, R13, R16, R17, R18, R4, R5, R12, and R9.

The total sailing distance of the voyage is 222.09 Nm summed up from individual segments. Each segment length is presented in

Table 4. Voyage segment lengths. Source: authors.

Segment	Distance (Nm)
R11–R8	13.17
R8–R9	2.9
R9–R12	2.47
R12–R13	14.96
R13–R16	9.83
R16–R17	13.06
R17–R18	34.59
R18–R4	13.86
R4–R5	36.5
R5–R12	67.35
R12–R9	2.47
R9–R10	10.93
Total	222.09

.

Table 5. Ship’s particular specifications. Source: authors.

Type	Tug/ASD AHTS
Length over all	33.34 m
Beam over all	10.00 m
Maximum draught	4.40 m
Displacement	839.19 t
Gross tonnage	429 t
Maine engines	2 × Caterpillar MAK 8M332
Total power	3200 kW
Maximum speed ahead	13 kts

provides a detailed overview of the specifications for the vessel, which is classified as a Tug/ASD AHTS type, and the layout can be seen in Figure 5. It has a total length of 33.34 m and a beam measuring 10.00 m. The maximum draught of the vessel is 4.40 m. In terms of its weight, the vessel has a displacement of 839.19 tons and a gross tonnage of 429 tons. Powering the vessel are two main engines, specifically the Caterpillar MAK 8M332 model, which together generate a total power of 3200 kW. With these specifications, the vessel can achieve a maximum speed ahead of 13 knots.

The operational metrics of the ship based on its throttle or control lever’s position are shown in

Table 6. BSV speed reference. Source: Authors.

Lever Position (%)	Speed (kts)	Consumption (L/h)
5	3.2	48.64
15	5.1	94.86
25	6.5	131.95
35	7.9	194.34
45	9	241.2
55	9.9	289.47
65	10.8	347.22
75	11.4	411.08
80	11.9	495.03
85	12.3	544.64
90	12.6	582.49
95	12.8	628.73
100	13.0	688.87

. The “Lever Position” indicates the engagement level of the ship’s engine, with 100% representing full power. As this position increases, the ship’s speed in calm waters, denoted in nautical miles per hour (knots), typically increases. The fuel consumption rate, shown in liters per hour, also escalates as the lever position and speed increases.

The segments of the route that had quasi-identical characteristics were merged; thus, for computing reasons, the following segments shall be considered: R11–R8 (13.17 Nm), R8–R17 (43.22 Nm), R17–R18 (34.59 Nm), R18–R5 (50.36 Nm), R5–R12 (67.35 Nm), and R12–R10 (13.40 Nm). The vessel has two stops at R18 and R5.

Ten scenarios were considered, taking into account the time limit to complete the voyage, weather conditions (sea and wind), and displacement (

Table 7. Voyage scenarios. Source: authors.

Scenario	Voyage Time Limit (h)	Weather Conditions	Displacement (Tons)
Scenario 1	18	Ship fully loaded with fuel, weather scale BN = 1	700.46
Scenario 2	18	Ship fully loaded with fuel, weather scale BN = 2	700.46
Scenario 3	20	Ship fully loaded with fuel, weather scale BN = 3	700.46
Scenario 4	24	Ship fully loaded with fuel, oil, provisions and 50 t of cargo on board, weather scale BN = 1	811.91
Scenario 5	24	Ship fully loaded with fuel, oil, provisions, and 50 t of cargo on board, weather scale BN = 2	811.91
Scenario 6	24	Ship fully loaded with fuel, oil, provisions, and 50 t of cargo on board, weather scale BN = 3	811.91
Scenario 7	22	Ship loaded to the maximum permissible draft, weather scale BN = 2	839.19
Scenario 8	22	Ship loaded to the maximum permissible draft, weather scale BN = 3	839.19
Scenario 9	19	Ship loaded with 50% fuel, weather scale BN = 2	743.42
Scenario 10	19	Ship loaded with 50% fuel, BN = 3	743.42

).

For each voyage segment, Kwon’s Formula (8) was used to calculate the involuntary reduction in speed due to weather conditions and the ship’s state, whether it is cargo laden or in ballast. The ship’s reference speed is its designed cruising speed of 13 knots. Decisions by the ship’s captain to voluntarily reduce the speed were not considered in this analysis. Such decisions typically arise from concerns about navigating safely in harsh weather or channels with dense traffic. According to voyage regulations, the ship can sail in conditions up to a Beaufort number (BN) of 3 and avoids routes that intersect with high-traffic channels. A time frame restriction was added due to the schedule of the supplier and the offshore oil rigs.

For each voyage leg, the percentage loss of speed at a fixed engine power was calculated. The referenced speed is the designed maximum speed of the vessel. Some of the speed reduction percentages are negative, which implies conditions under which the ship experiences a speed increase due to favorable conditions (e.g., wind from astern).

4. Results

The calculated optimal speeds for the 24 h voyage are detailed in

Table 8. Optimized speed choice for travel time T = 24 h. Source: authors.

Voyage	Engine Speed (kts)
Scenario	R11–R8	R8–R17	R17–R18	R18–R5	R5–R12	R12–R10
1	9.90	10.80	9.90	9.90	9.90	9.90
2	10.80	11.40	11.40	11.40	10.80	11.40
3	12.80	12.60	12.80	12.60	12.60	12.60
4	9.90	10.80	9.90	9.90	9.90	9.90
5	10.80	11.40	11.40	11.40	10.80	11.40
6	12.60	12.60	12.80	12.60	12.60	12.60
7	10.80	11.40	11.40	11.40	10.80	11.40
8	12.60	12.60	12.80	12.60	12.60	12.60
9	10.80	11.40	11.40	11.40	10.80	11.40
10	12.80	12.80	12.60	12.60	12.60	12.60

for each outlined scenario. The optimum lever position for the speed set-up is represented in Figure 6. The resulted lever position groups for the chosen scenarios are categorized based on shared attributes, like the BN. For the 32 h voyage, the speed values are presented in

Table 9. Optimized speed choice for travel time T = 32 h. Source: authors.

Voyage	Engine Speed (kts)
Scenario	R11–R8	R8–R17	R17–R18	R18–R5	R5–R12	R12–R10
1	6.50	6.50	7.90	6.50	9.00	7.90
2	9.00	9.00	9.00	6.50	9.00	9.00
3	9.90	9.00	9.00	9.90	9.00	9.00
4	6.50	6.50	7.90	6.50	9.00	7.90
5	9.00	9.00	9.00	6.50	9.00	9.00
6	9.90	9.00	9.00	9.00	9.90	9.00
7	9.00	9.00	9.00	6.50	9.00	9.00
8	9.90	9.00	9.00	9.00	9.90	9.00
9	9.00	9.00	9.00	6.50	9.00	9.00
10	9.90	9.00	9.00	9.00	9.90	9.00

, while the lever set-up for each group of scenarios is represented in Figure 7.

In Figure 6, we can observe the optimized speed choices for voyages with a travel time limit of 24 h. Figure 7 provides insight into the optimized lever choices, again, for a 24 h travel time limit. The lever position is directly related to the engine speed.

Figure 8 highlights the fuel consumption across the different proposed voyage scenarios and time limits. This figure summarizes the information from displayed in the earlier figure images, highlighting how the speed and lever choices affect fuel consumption. By using this comparison, the operators can identify the best fuel saving strategies for various voyage situations.

5. Discussion

The results underscore the trade-offs between speed and fuel consumption for two travel time limits: 24 h and 32 h. While higher speeds allow for quicker transits, they come at the cost of increased fuel consumption. The lever position directly influences the engine speed. Higher lever positions lead to higher engine speeds (Figure 6). For instance, in scenario 1, a lever position of 55% corresponds to an engine speed of 9.90 knots, while in scenario 3, a lever position of 95% corresponds to an engine speed of 12.80 knots due to ship speed variations because of the environmental condition. This emphasizes the critical decision making required in voyage planning: operators need to balance the urgency of the transit with the associated fuel costs.

When the travel time limit is extended to 32 h, as shown in Figure 7, we observe shifts in both the speed and lever choices. The extended time frame might allow for more flexibility in the speed choices, potentially enabling fuel savings without significantly compromising on the transit times. Similar to Figure 6 but for a 32 h limit, Figure 7 details the lever positions. The comparisons between the figures could reveal how lever choices adapt when given more time for the voyage.

The practical application of our method utilized real-world data for the vessel’s speed and fuel consumption, gathered under calm water conditions as presented in Table 6. To cater to the unique requirements of the voyage and prioritize operational safety, the weather conditions taken into account had a BN = 3. While there are various methods to compute speed losses under these conditions, we opted for a well-established and tested empirical method [26,27].

Numerous studies have explored the issue of speed optimization for ships engaged in long voyages. A vessel’s speed at sea is influenced by various factors, both voluntarily and involuntarily. Unlike other methods, our model provides a cost-effective solution tailored for short coastal journeys, such as those in the offshore supply context. Here, the primary factors affecting speed are typically weather conditions and constrained time frames.

6. Conclusions

This paper introduced a method for the optimization of speed choices using the backtracking algorithm for vessels in coastal voyages.

The proposed backtracking algorithm method can serve as a valuable tool for ship operators in the offshore industry. It provides a structured and efficient approach to voyage planning, enabling them to make informed decisions that optimize fuel consumption, reduce costs, and minimize environmental impacts.

It can be implemented as a low-cost evaluation tool for managers and stakeholders in the offshore industry to predict fuel consumption under different speed conditions and with different time restrictions.

Moreover, our research fills a significant gap in the maritime literature. While speed optimization has been extensively researched, the application of the backtracking algorithm to offshore supply vessels remains underexplored, making our study both innovative and of practical significance for industry stakeholders.

Future research can further enhance the algorithm by considering additional parameters and constraints specific to different vessel operations. Additionally, incorporating real-time data and advanced predictive modeling techniques can improve the accuracy and effectiveness of fuel optimization strategies. By doing so, we can ensure that the maritime industry moves towards more sustainable and cost-effective operations. Furthermore, a comparative study with other optimization techniques, real-time algorithm applications, and evaluations of its environmental impact, particularly with alternative fuels, can present a holistic view of its potential.