Optimization Model for Selection of the Offshore Fleet Structure

Abstract

Over the past century, with an accelerated increase in world energy demand, oil remained the most intriguing energy source, while oil exploration and production evolved from an onshore to an offshore exploration and production facility. In logistic and operational planning, the supply of offshore installations is the primary dependent of offshore supply vessels (OSVs). OSVs are classified into two main specialized groups, Anchor Handling Tug Supply (AHTS) and Platform Supply Vessels (PSVs). The establishment and maintenance of offshore oil installations require considerable financial resources for the hire of OSVs, which increases the oil exploration and production operator’s overall budget. Planning the offshore fleet structure aims to reduce expenses and increase the utilization of OSVs. This work will analyze the actual data of the southwestern African offshore market based on 24/7 working time and the use analysis of marine activities to define the parameters of the offshore fleet structure. The research results will be used to develop a methodology for modeling the optimal infrastructure of the offshore fleet. With statistical data of all support vessels’ activities and the use of linear programming (LP), it is possible to determine the vessel’s employment pattern and the optimal fleet structure by type and number of OSVs.

1. Introduction

Although oil has been discovered on land, over time it has been shown that oil fields also exist under the sea. For the first time in the world in 1846, oil was extracted by an industrial method from a well located 21 m deep in the Bibiehbat oil field in Azerbaijan [1], marking the beginning of a new era in the oil and gas industry.

Wells at the time were connected to the mainland by piers. It did not take long to replace the piers with free-standing platforms at sea supplied with offshore special-purpose vessels. The production and exploration of oil at increasing sea depths and in areas farther from the coast have led to the development and construction of modern OSVs of special characteristics that supply oil installations and perform other special tasks.

Such vessels have superstructures on the bow, while the stern part of the vessel is a large deck which in some types of OSVs has no railing at the stern to make it easier to manipulate anchors, tow, and position oil rigs (Figure 1 and Figure 2).

OSVs have their origins in the Gulf of Mexico, where oil exploration first moved offshore in the 1950s. World War II surplus vessels, wooden fishing boats, and shrimp trawlers were used to supply offshore platforms with cement, mud, spare parts, crew, fuel, and food. In 1955, Alden John Laborde developed the first purpose-built OSV with a bow wheelhouse and a stern cargo deck that became the standard for offshore support vessels [2]. The main features of OSVs are closely linked to the characteristics, specific location, and requirements of the platforms, regardless of whether they are exploration or production platforms.

The different water depths of the seas where drilling and oil exploration installations operate have also led to changes in the characteristics of oil platforms, and thus the characteristics of OSVs where they have evolved along with other parts of the offshore oil industry.

This paper presents a literature review and analyzes the functioning of the conceptual logistic model of offshore oil installations at sea and the position and type of OSVs in relation to location and job requirements.

Through the mathematical modeling of the structure of an offshore fleet, which includes the deployment and engagement of offshore vessels in the supply of offshore installations, the basic elements of planning the supply and defining the model of the supply of oil installations depend on the characteristics of offshore OSVs.

This research proposes a methodology for the selection of OSVs for the support and supply of offshore oil installations with the aim of reducing the total cost of fuel and the hire and port fees of OSVs, as well as increasing the degree of utilization. In addition, this paper deals with the basic elements of planning the supply of offshore installations and analyzes the data of the actual supply chain of offshore installations in the oil fields of West Africa, on the basis of which the assumptions and methodology of modeling the supply chain of OSVs are defined, and then this problem is solved using the LP method. After all, for two representative types of OSVs, the obtained solution was tested in order to obtain an insight into the changes that occurred after solving the problem, all in terms of reducing expenses, and at the very end, the results were compared with actual changes in the schedule of activities of OSVs. All data used in the analysis were extracted from official vessel logs, which were compiled for a period of one year and later reduced to a monthly average. The implementation of the mathematical model was carried out in the “R” environment. R is a programming language used for statistical calculations and data visualization. It offers a wide range of statistical methods such as linear and non-linear modeling, classical statistical tests, time series analysis, classification, clustering, graphical techniques, and is highly extensible. In this work, the modeling problem is solved with the standard syntax of linear programming using the GLPK (GNU Linear Programming Kit) package with the standard program “R” [3].

LP is a case of mathematical programming, where the objective function and constraints are linear. The LP problem consists of optimizing (minimizing or maximizing) the value of the linear objective function of the vector of decision variables given that the variables can only have values defined by a set of linear constraints. LP is one of the most frequently used techniques in the toolbox of quantitative optimizing methods, and it is also relatively simple and reliable for solving the challenges within this research.

2. Literature Review

The supply of offshore oil installations is a very challenging and demanding job that requires careful planning and an uninterrupted logistics chain in which vessels are a very important and crucial factor. There is little scientific research focused on the planning of the structure of an offshore vessel fleet. Most of the research in the offshore vessels industry is based on a vessel’s routing plan, and recently on human error while using a Dynamic Positioning (DP) system on an offshore vessel.

One of the first research investigations related to offshore vessel fleet modeling was made in 2000. Fagerholt & Lindstad [4] investigated the expenses and effectiveness of the supply of offshore installations and an optimal weekly routing plan for the assigned oil field with chartered offshore vessels in the North Sea.

Aas et al. [5] researched upstream logistics on the Norwegian continental shelf for the “optimal offshore vessel” and described OSVs as a crucial and the largest cost element in the offshore supply logistics chain. Total fleet capacity, sailing time, and loading and unloading capabilities are established as the main characteristics of supply vessels. In view of the total vessel capacity in the field, the voyage duration, and the time for cargo manipulation, the authors summarized the total vessel expenses for operators in the Norwegian Sea. Utilization of OSVs can be increased by using a better routing plan, but there is no mathematical model presented.

Shyshou et al. [6] dealt with the problem of fleet size in research platform relocation operations, where operations are not subject to a pre-made annual schedule but to the completion of the research activities of an individual oil platform. In this research, as with most authors who deal with this issue, the area of the Norwegian Sea is considered, while the focus of the research is the offshore installations of the Equinor ASA oil company (formerly StatoilHydro). They solve this problem by applying a simulation model, with the use of the ARENA program package. The stochasticity of the problem is defined by the duration of the transfer itself, as well as unfavorable weather conditions. In addition to the above, the problem that arises is the relocation of several exploration platforms at the same time, which requires several AHTS for handling anchors and towing. As the transfers take place without any predetermined order, the paper considered the use of AHTS that are on long-term lease, as well as those that must be hired for a short period. Offshore vessels hired for a shorter period of time are far more expensive than vessels hired for a longer period of time, so the impact on the budget that oil companies have for individual projects is also considered. The significance of the work is reflected in the results of the conducted simulation, where the authors state that the cost-optimal number of long-term vessel charters for the relocation of an oil platform is insensitive to the increase in the future price of short-term charters in the area of average to above average. They notice a greater sensitivity in the case of a decrease below the average. The results of the simulation model presented in the paper confirm operationally known facts, but only for one type of offshore activity and AHTS vessels.

In his master’s thesis, Rose [7] investigated the development of offshore vessels and the possibility of minimizing the total expenses of offshore vessels whose operational characteristics can meet all the requirements of the oil field and logistics. Assuming that the future development of the offshore industry will move to deeper water, the author compared the supply scenarios of oil installations in shallow and deep water. His scenario and mathematical calculations are based on supplying two drilling platforms and three production platforms. Through research, the author tried to predict the future trend of the characteristics of OSVs. Primarily, the design of OSVs is considered, which identifies the trend of existing vessels and the abilities of the modern offshore fleet. Part of the analysis carried out includes reports in the field of the offshore industry, while the other part focuses on gathering data on vessel design and identifying design trends using parametric analysis. For fleet modeling, Rose used the mixed integer linear program (MIP).

Maisiuk & Gribkovskaia [8] researched supply vessel planning problems in supplying offshore oil and gas installations. Research in this paper was based on a weekly routine supply of different offshore installations with precise dates and times of departure and visits to each location. The vessel capacity, offshore installation opening and closing hours, weather conditions, and cargo requirements are considered during the research and creation of the mathematical model.

The authors, Pantuso et al. [9], presented a literature review on the fleet size and mix problem in maritime transport. The authors noted that shipping companies work in an unpredictable environment and that the main strategic question is how to design an optimal fleet of ships. The objective of this research was to minimize the total cost of the fleet, including the routing problem. The paper analyzed the methodology for different types of shipping, such as passenger, oil, bulk, general, container, tugs, and barges, as well as OSVs. However, the basic solution they provide to the linear programming problem is very basic and difficult to apply in the offshore industry, which is quite unpredictable, unlike the shipping industry.

Halvorsen-Weare & Fagerholt [10] characterized OSVs as the largest cost drivers in the upstream logistics supply chain. Their work initially was initially performed as a project with Equinor ASA, formerly known as Statoil, a Norwegian state-owned multinational energy company. Optimization in OSV planning has been described in research with two mathematical models, Arc-flow and Voyage-based models. Weather conditions in the Norwegian continental shelf were considered, particularly during the winter period. Bad weather conditions, particularly high waves, have a direct impact on a vessel’s speed and offshore unloading/loading operations as well. Wave heights up to 4.5 m have a reduction in sailing speed, whereas heights more than 4.5 m have “waiting on weather” mode and standing by waiting for better weather.

Skoko et al. [11] in their research presented the main types of OSVs with their utilization. The structure of the offshore worldwide fleet is presented as well. The comparison of the fleet size by vessel type, in the past and nowadays, shows fewer vessels but increased gross tonnage due to a change in the design of the OSVs, which are now bigger than before as explorations move from shallow to deep water. Commercial strategy, contracting method, and dynamic position are described in their work, and SWOT analysis (Strengths/Weaknesses/Opportunities/Threats) is pointed out as an appropriate tool to assist the owner and charterer during their future offshore fleet business planning.

A classification is conducted by gross tonnage and bollard pull for the most important type of OSVs. AHTS has been presented by Boko et al. [12]. Bollard pull (BP) is presented as a measure of the pulling power of the AHTS and a reference for commercial aspects as well. The authors pointed out BP as one of the key performance indicators for AHTS vessels, and the value of BP defines the vessel’s functionality for a particular job and hires accordingly. The work conclusion is that the design of AHTS did not change much, but power and size are increasing due to market requirements and more sophisticated technology implemented.

Sopot & Gribkovskaia [13] researched a single-vehicle routing problem with the pickup and delivery of multiple commodities. This problem was noticed in offshore upstream logistics and is important for oil companies that are operating offshore. In this paper, mathematic formulation was presented, and a metaheuristics algorithm yielding non-Hamiltonian routes where offshore location may be visited once or twice was described. As the offshore industry is quite unpredictable, the problem arises when for any reason the OSV needs to return to an already visited location or if the difference in unloaded and loaded cargo is quite large and there is no place to load the total available cargo to return to the base.

Skoko et al. [14], in their work, presented the utilization of the two main types of offshore vessels, AHTS and PSV, and the connection of the crude oil price with a vessel’s daily rate. Research in this work was conducted to analyze the real sector data and the actual operational use of vessels in comparison with their monthly and annual expenses. The analysis showed that both types of vessels were almost 50% unused or waiting for orders. This leads to the conclusion that it is necessary to create a mathematical model to show the need for a particular type of vessel in the oil field.

In their research, Bolstad et al. [15] discussed the optimal fleet size and mix of vessels to support maintenance activities at one or more offshore wind farms. The significance of this work is that the research is based on the maintenance of several offshore wind farm locations, farms which are, in principle, carried out with offshore vessels, the same ones used in the oil and gas industry. The paper mentioned stopping the production of wind turbines due to breakdowns or maintenance, which results in lost earnings. It is certainly worth noting the duration of the contract for the vessels working on maintenance, as well as the weather conditions that directly affect the choice of fleet composition, which is very important to mention as a connecting factor between the oil and gas and wind farm industries.

A planning problem for supply vessels with multiple bases that arise in offshore oil and gas activities was analyzed by Shyshou [16]. The aim of the author’s research was to design cost-efficient periodic voyages and schedules for OSVs, as well as respective fleet configurations. The study was conducted specifically for Statoil, the largest Norwegian offshore oil and gas operator, who requested to research two supply bases with a defined voyage start date, time, and a sequence of installation visits. The results led to an overall smaller number of vessels needed to perform all the voyages and supply all locations. The problem arises in any disruption regarding, for example, weather conditions, cargo delays at the port of loading, or delays of any kind.

In contrast to the modeling of the standard merchant fleet, the offshore fleet is conditioned by a much larger number of limiting and unpredictable factors, which makes the solution to the optimality problem more complex.

When analyzing the applicability under real conditions, it becomes clear that the models from previous scientific research studies on the optimization of an offshore fleet have certain limitations. In previous models, an ideal deployment of vessel activities was assumed, or constraints were set, that reduced the complexity of modeling the structure of the offshore vessel fleet, or solutions were limited to only one type of vessel and one particular activity. The main limitation observed is that some of the proposed models do not take into account all the operational problems encountered in the oilfield and therefore provide incomplete results, while others do not show the implementation and experimental results of the model used. In addition, most of the modeling and research have been carried out for the North Sea areas with the conditions and requirements of the company in question, and it is not known what results the same models generate in different weather conditions and with other operators.

For this reason, research is being carried out and a model is being created to optimize the offshore fleet in the oil fields of a specific area, but which can also be used in any other location.

3. Offshore Supply Vessels

OSVs are crucial and indispensable factors in the offshore supply chain. In the offshore industry, with vessels whose primary purpose is to transport passengers and carry out crew change, there are two basic types of vessels that are structurally built to transport cargo and carry out specialized marine activities. Constructionally, the main characteristic of both types is that the superstructure of these vessels is positioned on the bow, and the cargo deck extends along the entire aft part. Nowadays, these vessels are very powerful and reliable, whereas most newly built vessels are equipped with a DP system.

The most common activities of PSVs (Figure 1) include the transport of supplies and personnel/employees from land to oil platforms and vice versa, the transport of cargo from the platform to the shore, and work near the oil platforms. Under the main deck, the PSV is equipped with tanks for the transportation of oil base mud, cement, diesel fuel, technical and drinking water, and chemicals used in subsea drilling. The design of the PSV provides a large and spacious deck space to enable the transport of as many types of cargo as possible. The vessel’s deck should be long enough to accommodate drill pipes which, although nominally 12 m long, can often be longer. Therefore, the length of the open space of such a vessel is usually 15–20 m and above. Today’s usual deadweight of the PSV fleet is the main criteria of categorization and can be divided into small PSVs under 2000 DWT, medium 2000–4000, and large 4000 and above [17]. This type of vessel is characterized by good maneuverability.

Anchor Handling Tug Supply Vessel

AHTS (Figure 2) occupy one of the leading positions in the offshore oil industry and, in general, are multipurpose. These vessels are equipped with special winches and steel ropes that help handle the anchors of oil platforms, barges, and production tankers, and they can perform towing and repositioning operations. AHTS also have deck cargo space and below-deck space for liquid and bulk cargoes. In this cargo characteristic, they are similar to PSVs.

AHTS have a higher daily hire because of the special equipment they are equipped with, which is used for the specialized jobs and activities already mentioned. Unlike PSV, the use of winches and steel ropes requires an open stern of the vessel through which the ropes pass. The previously mentioned specialized jobs are performed by connecting steel ropes to floating objects. AHTS vessels have more propulsion power than PSVs, as this power is primarily directed to the winches, i.e., the power used to pull floating units or anchors. The pulling power of winches is expressed through the bollard pull (BP) certificate and is recorded in metric tons or kilo Newtons (kN). The BP is defined as the static force exerted by a vessel at zero speed on a hawser [18].

4. Problem Description and Mathematical Model

4.1. Problem Description and Methodology

The problem described below is presented with the aim of optimizing the structure of the offshore fleet and will use data related to a certain oil field in West Africa related to purpose, size, distances, vessel speeds, capacity, etc. On the basis of the data thus obtained, an analysis of the actual state of time and total expenses will be performed, and then it will be compared with the data obtained from the mathematical model for two main types of offshore vessels, PSV and AHTS. The research will be based on the analysis of the past work of the fleet of offshore vessels in the oil field of the Republic of Angola, which is located about one hundred nautical miles from the logistics distribution center and consists of fourteen production platforms, one drilling platform, two Floating Production Storage Offload (FPSO) tankers, and three accommodation barges for the crew and workers working in the field. All data were collected from official vessel logs (logbook). During the research period, two PSV and three AHTS vessels were employed in this oil field.

There are several factors that need to be analyzed to successfully define the structure of the offshore fleet. The price of oil and daily vessel hire are two main elements that create a framework for the implementation of projects to optimize the rate of offshore vessel hire. The activities performed by offshore vessels can be divided into navigation, activities in the port, supply of offshore facilities, maritime activities such as towing, rig move, anchor handling, etc., and time spent on standing by which can be the result of all waits, such as waiting at the anchorage of the field and in the port and all other waiting times in the field. In addition to these elements, there are others that also affect the execution of the project, such as port fees, mobilization and demobilization fees, fuel consumption and the necessary amount of cargo that needs to be transported/loaded/unloaded in ports, the amount of cargo that is unloaded/loaded on oil platforms, etc. By applying the presented methodology, the optimal structure of the fleet of offshore vessels is determined based on the vessel’s activities. The creation of a model for determining the structure of the fleet of offshore vessels envisages the minimization of the total expenses of the fleet and the maximum utilization of the fleet with two basic groups of offshore vessels while fulfilling all the requirements of the oil field operation. The results of the model will be used further to compare the expenses obtained on the basis of log records extracted from the vessel’s logs and to develop a methodology for monitoring and optimizing these processes. The correction of the process will be made based on the hire rate of the vessels and the facilities they serve. The goal of the correction goes in the direction of profitability and safety, that is the stability of the oil field supply process.

The fundamental document in which all the vessel’s activities and events on board are recorded is the vessel’s logbook. The log record in the vessel’s logbook is a daily record of the vessel’s master and officers on the bridge about all the activities and state of the vessel during the day. For the analysis of the log records, Excel software (Microsoft Excel 2019 MSO (16.0.10405.20015) 64-bit) tools were used, and to obtain the optimal solution of the set linear programming model, the R “open source” environment through the package “library (lpSolve)” was used, as well as other functions through the CPLEX optimization solver, which gave identical results.

4.2. Data Used

During a year research period, port expenses are calculated on a daily basis depending on the length of the vessel and amount to USD 7.5 per meter per day. As the average length of PSV is 73.6 m and AHTS 68.95 m, the daily port expenses of these vessels are USD 552 and USD 517.13 per day. The average daily hire of a PSV is USD 27,500, while the average daily hire of an AHTS is USD 32,000 for the oil field analyzed. The average fuel price per ton is USD 870. All these data are input variables. To create a fleet optimization model, the annual data for two typical offshore vessels—PSVs and AHTS—were reduced to monthly averages. These two typical vessel types carried out the activities for which they were chartered, such as sailing, loading/discharging cargo in ports, supplying oil platforms, anchor handling, towing, waiting on platforms, etc.

Table 1. The annual data reduced to the average monthly operational expenses of PSV.

PSV	Port	Sailing	St/by Facility	St/by Anchor	St/by Port	Offshore Supply
Fuel consumption t/h	0.03	0.5	0.03	0.03	0.03	0.5
Daily fuel expenses	USD 626.4	USD 10,440.00	USD 626.4	USD 626.4	USD 626.4	USD 10,440.00
An average monthly activity time (day)	4.39	7.28	0.36	6.63	4.79	3.66
Total fuel expenses per activity (USD)	USD 2751.03	USD 76,018.00	USD 849.20	USD 4151.01	USD 2998.72	USD 38,213.72
Total fuel expenses all activities (USD)			USD 124,981.68
Daily port fee: 73.6 m × USD 7.5			Length over all = 73.6 m port fee per meter of the lengths = USD 7.5 the number of paid port fees (frequency of entry per month) = 8 73.6 × USD 7.5 × 8 = USD 4416.00
Daily hire: USD 27.500			29 × USD 27,500 = USD 797,500.00
Total expenses			USD 124,981.68 + 4416.00 + 797,500.00 = USD 926,897.68

shows the actual average monthly expenses through one-year data analysis of a typical PSV in terms of fuel consumption, daily hire, average time spent performing specific activities, estimated port fee, and finally the total shown at the very end of the table. The highest monthly expense of a PSV is the daily hire, which amounts to 86% of the total monthly expenses. The vessel was almost half of the hire period standing by waiting for orders.

Table 2. The annual data reduced to the average monthly operational expenses of AHTS.

AHTS	Marine Activities	Port	Sailing	St/by Facility	St/by Anchor	St/by Port	Offshore Supply
Fuel consumption t/h	0.5	0.03	0.5	0.03	0.03	0.03	0.5
Daily fuel expenses	USD 10,440.00	USD 626.4	USD 10,440.00	USD 626.4	USD 626.4	USD 626.4	USD 10,440.00
An average monthly activity time (day)	5.29	2.11	5.96	2.94	7.73	2.36	2.58
Total fuel cost per activity (USD)	USD 55,211.77	USD 1322.76	USD 62,243.06	USD 1840.23	USD 4841.33	USD 1480.52	USD 26,929.52
Total fuel expenses all activities (USD)				USD 153,869.20
Daily port fee: 68.95 m × USD 7.5				Length over all = 68.95 m port fee per meter of the lengths = USD 7.5 the number of paid port fees (frequency of entry per month) = 8 68.95 m × USD 7.5 × 8 = USD 4137.00
Daily hire: USD 32.000				29 × USD 32,000 = USD 928,000.00
Total expenses				USD 153,869.20 + 4137.00 + 928,000.00 = USD 1,086,006.20

shows the actual average monthly expenses through one-year data analysis of a typical AHTS vessel. The difference between PSV and AHTS vessels is that this type, unlike PSVs, and the equipment they own, carries out maritime activities such are anchor handling, towing, tanker lifting, rig move, etc.

Table 3. The relationship between the annual data reduced to the average monthly expenses of PSV and AHTS vessel.

Cost/Vessel	Fuel	Port Fee	Monthly Hire	Total
PSV	USD 124,981.68	USD 4416.00	USD 797,500.00	USD 926,897.68
AHTS	USD 153,869.20	USD 4137.00	USD 928,000.00	USD 1,086,006.20
				USD 2,012,903.88

compares the actual cost ratio of an average PSV and AHTS vessel for fuel consumption, port fees, and hire during a year-long research period.

It can be seen that the hire amount is the largest item in the total average monthly expenses of employing offshore vessels. The total monthly expenses of this arrangement amount to over USD 2 million for two vessels only.

4.3. Mathematical Model Formulation

Work analysis and mathematical modeling of the optimization of expenses and the budgeting of optimal times of offshore vessels in the researched oil field aim to confirm that it is possible to significantly reduce the total expenses of the engagement of the fleet of OSVs that increase the degree of their utilization during the time when the vessels are on hire and redirect PSVs whose average daily hire is lower than an AHTS in performing all those activities that this type of vessel can perform as efficiently as AHTS. On the other hand, the mathematical model aims to confirm that it is possible to reduce the hire time of an AHTS vessel by almost half a month, and its mandatory hire only in the performance of maritime activities, even at the price of hiring an AHTS on the short-term vessel market, for which the daily rate is higher than during a long-term contract [2].

The general elements of analysis and planning (

Table 4. The annual data reduced to the average monthly input data for two types of offshore vessels.

Input Data	PSV	AHTS
Average economic speed (knot)	10	11
Average vessel capacity (t) Usable carrying capacity 85% of the average capacity of the vessel (t)	1600 1360 cargo + bulk + mud	800 680 cargo + bulk + mud
Average vessel length (m)	73.6	68.95
Average daily port fee (USD/day)	552	517.13
Average hourly/daily fuel consumption—in the performance of maritime activities (t/h/t/day)		0.5/12
Average hourly/daily fuel consumption—loading/unloading cargo in the port (t/h/t/day)	0.03/0.072	0.03/0.072
Average hourly/daily fuel consumption—in navigation (t/h/t/day)	0.5/12	0.5/12
Average hourly/daily fuel consumption—in supply operations (t/h/t/day)	0.5/12	0.5/12
Average hourly/daily fuel consumption—waiting—platform (t/h/t/day)	0.03/0.072	0.03/0.072
Average hourly/daily fuel consumption—waiting—port (t/h/t/day)	0.03/0.072	0.03/0.072
Average fuel consumption per hour—waiting—anchor (t/h/t/day)	0.03/0.072	0.03/0.072
Average fuel price (USD/t)	870	870
Average number of days per month (day)	30	30
Average daily loading/unloading of cargo in the port (t/day)—obtained from data from the vessel’s logbooks (cargo manifest)	2478	680
Average daily loading/unloading of cargo on the platform (t/day)—obtained from data from the vessel’s logbooks (cargo manifest)	2972	557
Average daily hire (USD/day)	27,500	32,000
Total average (monthly) distance in nautical miles sailed by vessels (NMs)	3322
Minimum average (monthly) amount of cargo that (PSV and AHTS) vessels unload when supplying platforms (t/month)	12,316
Minimum average (monthly) amount of cargo that (PSV and AHTS) vessels unload when supplying platforms (t/month)	12,316
Minimum average (monthly) amount of cargo loaded/unloaded in the port and loaded/unloaded on the platform (t/month)	21,760
Average fuel price (USD/t)	870

) that will be used in the research and the presented mathematical model are the loading and unloading characteristics of the LDC and offshore installations and the characteristics of the offshore vessel.

According to these inputs, an optimization model will be set up with the following assumptions:

• Each offshore vessel departs the port loaded with cargo, sails to the oil field, unloads/loads the cargo, and then returns to the port where it unloads/loads the cargo, and returns to the oil field again where it touches several oil platforms that it supplies (loads/unloads) necessary materials, spare parts, cargo, etc.;
• The needs of the platform are shown as the sum of the requirements of the total amount of cargo;
• The economic speed of an offshore PSV is 10 knots, while that of an AHTS vessel is 11 knots;
• Total vessel expenses include vessel hire, port fee, and fuel for individual types of activities (boarding/disembarkation, waiting, sailing and maritime activities).

The mathematical model of the LP optimization of the supply cycle of oil platforms has the following:

1. The objective function, which should be minimized (given that it is about expenses) and which depends on the product of expenses per unit of activity/operation and time (x_k,i—number of hours of performance of the i—activity by the k—vessel);

2. A set of constraints that represent a linear combination of variables with coefficients marked most often with the symbol a_ij must be less than, equal to, or greater than the right side of the limit, usually marked with b_i.

Modeling and problem solving are conditioned by the precise definition of constraints and the objective function that represents the total expenses (fuel, port expenses, and hire).

The model set up in this way is used in the context of improving the current state of the employment of offshore vessels based on the analysis of real data obtained from vessel logs. The goal of the model defined in this way is to minimize the operating expenses of coastal vessels and define the optimal structure of the fleet.

Thus, the formulation of the problem using the LP method in a canonical form, whose task is to minimize the expenses of planning the time of certain operations/activities of PSV and AHTS vessels, such as cargo loading in the port, maritime activities, navigation, supply, and standing by, can be presented as follows:

Min over $x,\alpha ,\beta$

$$\sum _{k=1}^2\sum _{i=1}^m{G}_{k,i}{x}_{k,i}+\sum _{k=1}^2{C}_k{\alpha }_k+\sum _{k=1}^2{F}_k{\beta }_k$$

(1)

In the set formulation, the decision variables through which the minimum value of the objective function is sought are:

(a)

${ x}_{k,i}$—continuous variable of the time consumption of the k vessel in performing the i activity;

(b)

α_k—an integer variable that represents the number of hire days of the k vessel (k = 1 is PSV, k = 2 AHTS vessel);

(c)

β_k—an integer variable that represents the number of days of use of port services for the k vessel (k = 1 is PSV, k = 2 AHTS vessel).

Example: if $x_{\mathrm{2,1}}$ = 0.5, where 2 (k = 2) represents the AHTS vessel, while (i = 1) represents maritime activities, this means that the AHTS vessel spends half of its working day conducting maritime activities.

Thus, the first term in the object function in Equation (1) represents the total fuel expenses (G_k,i) for each i activity performed by the k type of vessel, i.e., PSV (k = 1) and AHTS (k = 2).

The second expression (C_k) represents the daily hire expenses of the k vessel, while (F_k) represents the port fees for the k vessel per day.

The coefficient C_k represents the daily expenses of hiring a PSV (k = 1 and amounting to USD 27,500 per day) and for an AHTS (k = 2 and amounting to USD 32,000 per day).

Daily port charges (F_k) are paid at USD 7.5 per meter of vessel length per day. The average PSV is 73.6 m long, so the daily port expenses are USD 552; that is, the AHTS is 68.95 m long, so the daily port expenses are USD 517.13.

All data for known elements of the objective function and constraints are presented in detail in Table 4.

(a)

Set m includes all the maritime activities of the k vessel, which are indexed by i in the formulation of the problem;

(b)

Set K includes all available vessels and is indexed by k;

(c)

D maximum number of days in a month of engagement of a certain vessel (up to 30);

(d)

Set T includes the smallest average amount of cargo that is loaded/unloaded in the port or loaded/unloaded on platforms on a monthly basis;

(e)

Set W includes the minimum average amount of cargo loaded/unloaded, whether it is loaded/unloaded in the port or loaded/unloaded on the platforms, while the minimum average amount of cargo loaded/unloaded in the port and on the platform is marked with T. As it is known based on the log record that the minimum average amount of cargo handled in the port is greater than the minimum average amount of cargo handled on the platform, the larger of these two average values was taken in the mathematical model. This means that neither in the port nor on the platform can the minimum amount of handled cargo be less than this value.

The set of constraints, for the set objective function (1), is displayed as follows:

$$\sum _{i=1}^m{x}_{k,i}\le D_k, \forall k$$

(2)

where:

• D_k—the maximum number of hire days of k—vessel in the performance of all i—activities, maximum 30.

$$\sum _{k=1}^2{t_{k,i}x}_{k,i}\ge T$$

(3)

where:

• $t_{k,i}$—average speed of cargo loading/unloading in the port k—vessel per day (t/day);
• i—activity of cargo loading/unloading in the port;
• T—average minimum amount of cargo loaded/unloaded in the port (t).

$$\sum _{k=1}^2{w_{k,i}x}_{k,i}\ge T$$

(4)

where:

• $w_{k,i}$—average speed of cargo loading/unloading on the platform for k—vessel per day (t/day);
• i—activity of cargo loading/unloading on the platform;
• T—average minimum amount of cargo loaded/unloaded on the platform (t).

$$\sum _{k=1}^2{t_{k,i}x}_{k,i}+\sum _{k=1}^2{w_{k,j}x}_{k,j}\ge W$$

(5)

where:

• $t_{k,i}$—average speed of cargo loading/unloading in the port per day;
• $w_{k,i}$—average speed of cargo loading/unloading on the platform per day;
• i—activity of cargo loading in the port;
• j—activity of cargo unloading on the platform;
• W—the smallest average amount of cargo loaded/unloaded in the port + the amount of cargo loaded/unloaded on the platforms.

$$\sum _{k=1}^2{s_{k,i}x}_{k,i}\ge S$$

(6)

where:

• $s_{k,i}$—distance with economic sailing speed for k—vessel per day (NMs/day);
• i—sailing activity for k—vessel;
• S—the total distance at sea that the vessels have to sail (NMs).

$$\sum _{i=1}^m{x}_{k,i}\le {\alpha }_k, \forall k$$

(7)

where:

• x_k,i—time (days) of the k—vessel in all the activities it performs.

The sum of time must be less than or equal to the number of vessel hire days

$$x_{k,i}\le {\beta }_k, \forall k$$

(8)

where:

• x_k,i—the time of the k—vessel in the port must be less than or equal to the number of days for which port dues have been paid.

$${\alpha }_k—\mathrm{whole} \mathrm{number},\forall k$$

(9)

$${\beta }_k—\mathrm{whole} \mathrm{number},\forall k$$

where:

• the number of hire days (α) and the number of port dues payment days (β) for the k—vessel are whole numbers (days). Hire is paid daily.

$$x_{k,i}\ge \mathrm{2,4}, \forall k$$

(10)

• i—activity waiting at the port of the k vessel.

$$\mathrm{2,4}\le \sum _{k=1}^2{x}_{k,i}\le 5$$

(11)

• x_k,i—the waiting time (except waiting in port) for both vessels is defined by a lower and an upper limit. Thus, the mathematical model defined that the minimum sum of all waiting times, except for waiting in the port, for PSV and AHTS together cannot be less than 2.4, nor more than 5 days.

$${M_1\le x}_{2,i}\le M_2$$

(12)

• x_2,i—the time of carrying out maritime activities for an AHTS can be defined by lower and upper limits. M1 represents the lower limit in days, while M2 represents the upper limit in days of AHTS hire.

$${\beta }_k\ge minimum, \forall k$$

(13)

• ${\beta }_k$—the minimum number of days a vessel can use port services is 8 for a PSV, while it is 7 days for an AHTS.

$$x_{k,i}\ge 0 \forall k$$

(14)

Constraints and assumptions play a critical role in the selection of offshore fleet vessels. Therefore, it is important to accurately understand the additional assumptions and constraints that are introduced into the cost and time optimization model of individual activities. PSVs and AHTS can be chartered for a certain number of days (whole numbers) and spend a whole number of days in the port because port fees, depending on the length of the vessel, are paid for the whole day regardless of the number of entries and hours spent in the port in one day. Vessels can enter and leave the port several times during one day.

All other activities such as the number of days in navigation, supply, waiting at anchor, waiting in port, etc., must be equal to or greater than zero, but they do not have to have an integer value and represent time in days.

The constraint in Equation (2) ensures that the sum of all the time spent in all activities performed by an individual vessel (PSV or AHTS) is less than or equal to the number of days in a month, i.e., the maximum utilization of a certain vessel per month (maximum 30/31 depending on the number of days of the month).

The minimum average amount of cargo handled by both vessels in the port (T₁) and on the platform (T₂) is not the same, and this difference is due to the fact that the precise weights of all the handled cargo are not recorded in the vessel’s logbooks, but approximate cargo weights are recorded. Considering that T₁ is slightly higher than the value of T₂, in the mathematical model, a higher value was taken and entered only once (T). Therefore, T cannot be less than the minimum average value of handled cargo for both vessels (in port or on platform). The actual value of T ranges slightly more than half of the value of W (as can be seen in Table 3).

The constraint in Equation (3) ensures that the total amount of loaded/unloaded cargo (T) in the port for both vessels cannot be less than the minimum average amount of handled cargo in the port (T) for both vessels, taking into account the average daily rate of the loading/landing of the k vessel (t_k,i) in the port.

The constraint in Equation (4) ensures that the total amount of loaded/unloaded cargo on the platforms (T) for both vessels cannot be less than the minimum amount of handled cargo on the platforms, taking into account the average daily speed of loading/unloading of the k vessel (w_k,i) on the platforms.

Due to cargo handling and safety reasons, vessels are loaded in the port up to a maximum of 85% of their carrying capacity. The average PSV has a carrying capacity of 1600 t, so a capacity of 1360 t is taken, and an average AHTS has a carrying capacity of 800 t, so the carrying capacity is taken as 680 t. All average data for PSV and AHTS are given in Table 4.

Equation (5) defines the minimum monthly total amount of cargo handled by both vessels together W in the port and on platforms.

It is logical that all the loaded cargo in the port must be unloaded on the platforms and vice versa. The loading/unloading speed in the port and platforms for each of the vessels is not the same. Thus,${ t}_{k,i}$ represents the average speed of loading/unloading of cargo in the port per day, while $w_{k,i}$ represents the average speed of loading/unloading of cargo per day in supply points/platforms for the k vessel.

The constraint in Equation (6) ensures that vessels must spend a certain number of days in navigation and sail a minimum distance (S—the minimum distance in nautical miles (NMs)) for the observed time period, i.e., month). The average economic speeds of a typical PSV and AHTS are 10 and 11 knots (NMs/hour), respectively. So, the average daily distance covered by a PSV is 240 NMs, and for an AHTS, it is 264 NMs. The total minimum average monthly distance that must be sailed by both vessels is known (S) and can be seen in Table 4.

Equation (7) limits the total fraction of time that an individual vessel uses in performing all activities. This sum must not exceed the number of days it is possible to hire the vessel, which means a maximum of 30.

Equation (8) limits the total time of an individual vessel in port activities, i.e., loading/unloading cargo or port access. Considering that port dues are paid by the length of the vessel and per day, the total time spent for the k vessel in the port must be less than or equal to the number of days for which port dues are paid. Payment of port fees is done on a daily basis, and a vessel for which port fees have been paid can access the port for loading/unloading cargo in one day without paying additional expenses.

The number of days in port, as well as the number of charter days for the k vessel, should be integers, as defined by constraint (9).

As the goal is to reduce the lost time, i.e., waiting for the k vessel to the minimum possible extent, constraint (10) defines the lower limit of waiting in the port given that only waiting in the port is independent of the desire and ability of the vessel operator and cannot be less than 8% of the month, or 2.4 days per month.

Considering that the average PSV spends about 4 days in the port, and on average visits the port about 32 times, it is defined that the maximum number of days spent in the port for a PSV can be 8 days, while an AHTS spends a total of about 2.11 days in the port and visits the port about 16 times. For the AHTS, the upper time limit for the stay of the AHTS in the port is defined as 7 days.

The length of all waiting times (except waiting at the port) is defined in Equation (11). Thus, the minimum sum of all waiting times for PSV and AHTS is a minimum of 2.4 days, while the upper limit of all waiting times for both vessels is limited to a maximum of 5 days. These data were obtained after the analysis of real data, and the upper limit was defined based on them.

Equation (12) limits the minimum and maximum number of hire days of a certain vessel (for example, AHTS). If there is a possibility of eliminating the AHTS from the engagement, i.e., excluding it from the fleet, due to its inefficiency or significantly higher total expenses, compared to the more economical engagement of the PSV, this limitation makes it impossible since the oil field must not be left without at least one AHTA, for safety reasons, that undertakes the performance of all maritime activities for which there may or may not be a need at a certain time in the oil field. This limit is used to define the minimum and maximum time period of the possible engagement of the AHTS, so it is defined that the AHTS hire cannot be less than 7 but not more than 15 days per month based on the analysis of real data and based on experience in the profession.

Based on the data from the vessel’s logs, it was observed that the average usage of the AHTS on a monthly basis does not exceed 15 days.

The defined constraints are very flexible; it is possible to easily modify and adapt them to larger oil fields, with a larger and smaller number of offshore vessels, as well as other types of vessels, etc. This also implies that the mathematical model is applicable with minor modifications to oil fields all over the world, and not only for the area of Angola.

The minimum number of days a vessel needs to use port services, which is 8 for PSV and 7 days for AHTS, is defined by Equation (13).

According to the analyzed data, it is evident that the minimum number of days the vessel spent in port for PSV is 8, while for AHTS it is 7 days.

Constraint (14) is a natural constraint, ensuring that times cannot be negative numbers.

5. Data Analysis and Results

The implementation of the mathematical model was carried out in the “R” environment after the formulation of the mathematical model was transformed into a standard form using the functions defined in the R software package(R version 3.4.1 (2017-06-30) – “Single Candle”) to obtain solutions of variable values that represent the times of performing certain activities of PSV and AHTS vessels and to connect unknown values with an objective function representing optimal minimum expenses, respectively. Based on actual data on engagement time and according to the types of operations/activities, actual expenses were calculated for the selected PSV and AHTS, and an LP mathematical model was generated that determines the optimal engagement time and expenses of the set model. Through preliminary analysis of the real data, on the basis of which the mathematical problem was generated, it was solved by the LP method. On the basis of this solution, the optimal pace of activities, a reduction in waiting time for vessels, increased utilization, and reduced total expenses were determined. Table 4 shows the optimal times in the day for each PSV and AHTS engagement activity. It can be seen that the AHTS will no longer perform port services, as well as supply activities, since it will be taken over by the PSV at the time when it would be on standby. Waiting at the anchor for both vessels has been significantly reduced, while waiting in the port has been reduced to a minimum of 2.4 days per month. This means, in effect, that the AHTS will mainly carry out navigation activities and not spend time loading cargo in the port as these activities will be taken over by the PSV. The AHTS remains available for 2.4 days for port operations, which are visible in

Table 5. Optimal engagement times of PSV and AHTS by activity.

Optimum Time (Day)	Maritime Activities	Port	Navigation	St/by Platform	St/by Anchor	St/by Port	Supply
PSV	0.00	3.81	13.18	2.40	0.00	2.40	4.14
AHTS	7.00	0.00	0.60	0.00	0.00	2.40	0.00

as port waiting time. Since the waiting time in the port cannot be influenced, at least 2.4 days are left available, as previously defined in the restrictions.

Nevertheless, the use of port services and the payment of port fees for a minimum of 8 days for a PSV are foreseen, although the sum of the time that the PSV spends in the port is reduced to 3.81 days, while the time that the AHTS needs to spend in the port is reduced to 0, but port charges for at least 7 days have been paid for AHTS.

Table 6. Optimal hire days.

Vessel	Daily Hire (USD)	Hire (Days)	Port (Days)	Daily Port Fee (USD)
PSV	USD 27,500.00	26	8	USD 552.00
AHTS	USD 32,000.00	10	7	USD 517.13

shows the number of hire days for both ships together with the number of days of using port services. The number of hire days for the AHTS ship decreased from 29 (as it was in reality) to only 10 days. This means that it is possible to reduce the number of AHTS vessels in a selected oil field in the Republic of Angola from three to one and still carry out all activities.

At the same time, the PSV ship was not used for all thirty days but rather only for twenty-six days, which means that every month there is the possibility of its engagement for another four days in all activities for which the need is shown in the oil field, especially since ships are rented for a longer period of time and capable of performing all other activities except maritime ones, for which the AHTS ship is specialized.

Table 7. Optimal expenses.

Optimal Expenses	Optimal Fuel Consumption (USD)	Optimal Port Fee (USD)	Optimal Hire (USD)	Total Optimal Expenses (USD)
PSV	USD 186,274.10	USD 4416.00	USD 715,000.00	USD 905,690.10
AHTS	USD 80,847.36	USD 3619.88	USD 320,000.00	USD 404,467.24

shows the optimal expenses by type of vessel. It can be seen that the total expenses of hiring the AHTS have decreased significantly and amount to only USD 404,467.24. Expenses are significantly reduced for the AHTS and slightly less for the PSV. The expenses of hiring an AHTS were reduced the most, by more than 50%, compared to the actual expenses previously incurred.

Table 8. Comparison of optimal versus actual expenses.

Expenses	Optimal/Minimum Expenses (USD)	Actual Expenses
PSV	USD 905,690.10	USD 926,897.68
AHTS	USD 404,467.24	USD 1,086,006.20

shows a significant reduction in total expenses, especially for AHTS. With this solution, the expenses obtained by the mathematical optimization model, in relation to the actual expenses, amount to 65%, which means that there was a reduction in the total expenses for the PSV and AHTS by 35%, or USD 702,746.55 per month.

6. Conclusions

The results of the success of oil installations depend on the quality and efficiency of the established supply chain. The continuity of the supply chain is ensured by the proper planning of the number, type, and usage of OSVs, and the characteristics of offshore supply vessels are closely related to the sea distances over which the rigs operate regardless of whether they are exploration or production platforms. In addition to all that, the price of crude oil has a great influence on the world’s economic trends, including the daily hire prices of OSVs as well as the construction of exploration platforms. Given that oil production is an expensive process, and OSVs are an integral part of that process, the future development of technology in the oil industry will certainly have an impact on the development of OSVs and exploratory oil installations for deep seas in terms of capacity, length, and speed, as well as their maneuverability abilities. By choosing the “optimal” vessel, the number of vessels in the fleet, and the frequency of sailing limitations, such as port and oil rig congestion, weather conditions, vessel’s maneuverability, etc., respectively, a fleet that could operationally and efficiently serve one or more oil fields can greatly help not only operators but also owners of OSVs, as well as the entire offshore oil industry in the future planning of new oil fields and projects related to them. The research in this paper was carried out on West Africa oil fields and was focused on two main types of offshore vessels, PSV and AHTS. Based on actual data on engagement time and according to the types of operations/activities, actual expenses were calculated for the selected PSV and AHTS vessels during a year period reduced to the average month, and a linear programming mathematical model was generated that determines the optimal engagement time and expenses of the set mode. The goal of the research was to take a representative PSV and AHTS vessel with certain characteristics and make a model that would minimize expenses and determine optimal times for each of the activities of vessels employed in the oil field while meeting the supply needs of offshore installations. Since the supply of offshore oil fields has become more complex than the supply of only one platform, a mixed integer linear programming model of the fleet selection process has been developed and implemented on representative examples of offshore oil exploration and production for one of the fields in the Republic of Angola. The presented mathematical model gives an answer for the structure of oil field vessels, and this can be used to evaluate the profitability of representative vessels and the value of flexibility in terms of the design of OSVs.

This research proved that it is possible to determine the employment pattern of vessels through the application of the linear programming model of vessel activities and, according to it, define the optimal structure of the fleet by type (PSV and AHTS) and number of OSVs, namely two PSVs plus one AHTS instead of two PSVs and three AHTS ships for the researched field, in which maximum employment and minimum expenses of hiring vessels to service the oil field are ensured. The changes that the model clearly indicated are reflected in the reduction in waiting time, the increase in the degree of utilization of PSV and AHTS, the reduction in the expenses of hiring for both vessels, the reduction in the number of days of engagement of the AHTS, the need to reduce the number of AHTS serving the oil field, etc. In the end, the operators still have to decide how many and which vessels they will hire, for what period of time and under what conditions, and whether they will make decisions based on data that can be obtained based on estimates from appropriate mathematical models or on the basis of their own personal assessments.

The presented setting of the mathematical model with modified input parameters, depending on the size of the oil field and distances, is applicable to all other oil fields in the world.