Master’s Thesis Operations Research

Optimizing long-term maintenanceplanning for offshore gas installations

Eline Folkers

March 31, 2021

Supervisor: Prof. dr. R.H. TeunterSecond assessor: Dr. A.H. Schrotenboer

External supervisor: B. Groothuis

Master’s Thesis Operations ResearchSupervisor: Prof. dr. R.H. Teunter

Second assessor: Dr. A.H. SchrotenboerExternal supervisor: B. Groothuis

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Optimizing long-term maintenance planning for offshore gasinstallations

Author: Eline Folkers

Abstract

We consider the Fleet Composition and Deployment problem (FCDP) for offshore gas installations,where an optimal long-term fleet composition of vessels and helicopters for maintaining gas instal-lations at sea has to be determined. The work originates from a project with the NAM, a largeDutch gas exploration and production company. We propose a three-step deterministic modelingapproach. First, we strategically decide what transport types to use for both preventive and cor-rective maintenance operations, and we tactically decide how many days to deploy each transporttype such that all maintenance tasks can be completed within the planning horizon. Second, wedetermine the routing of vessels and helicopters for planned preventive maintenance tasks, giventhe cost-minimizing fleet composition from the first step. Third, we assign unexpected correctivemaintenance tasks to the fleet after simulating a Poisson process for each gas installation individu-ally. The idea of this methodology is that we predict the optimal strategic and tactical solution forthis stochastic problem and evaluate it on an operational level afterwards by observing the fractionof completed tasks. Our models are successfully implemented and tested on a real size FCDP. Wefind that climate transitions and a change in maintenance demand have a large effect on the optimalfleet composition and corresponding cost. Our models provide decision makers insights into the costand performance of fleet compositions under different parameter settings, enabling them to makestrategic decisions that minimize maintenance costs on the long-term.

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Preface

This thesis is the final product for obtaining the MSc. degree in Econometrics and OperationsResearch at the University of Groningen. After studying for more than six years at both the de-partment of Science and Engineering and Economics and Business, my academic career has cometo an end. I look forward to applying my knowledge of mathematics and econometrics into myfuture jobs.

Writing this thesis was a rather lonely job, especially with the pandemic going on. Luckily, I re-ceived a lot of help and support, and I want to use this preface to thank everyone who supportedme in this process and contributed to the completion of my thesis.

In the first place, I would like to thank Ruud Teunter for being my first supervisor. Thanks to yourinput and feedback, the quality of this thesis has improved a lot. I want to thank you for supportingme in my work and for always responding quickly to my emails. Secondly, I would like to expressmy gratitude to Albert Schrotenboer who supported me beyond his role of second assessor. Yourknowledge of the subject really improved this research and I would like to thank you for taking thetime to read my work and for participating in our meetings.

I had the pleasure to write this thesis in combination with an internship at ORTEC B.V. Specialthanks goes to my supervisor at this organization: Bas Groothuis. I want to thank you for guidingand motivating me on this project. Thank you for all the energetic brainstorming sessions we had,and for always keeping an overview of the bigger picture. I would like to thank everyone at ORTECwho has contributed to this project. It was nice working with such enthusiastic and intelligentpeople. In particular, I want to thank Bolor Jargalsaikhan for the great knowledge-sharing sessionswe had. Your enthusiasm and expertise on the subject gave me a lot of energy.

The assignment of this thesis project was commissioned by the NAM. I want to thank Jiri vanStraelen for the pleasant cooperation, for his understanding of the complexity of the problem, andfor thinking along about how to simplify the problem while still managing to reflect the practicalissues involved.

Finally, I want to thank my husband, family, friends, fellow students and sport buddies for sup-porting me during the writing of this thesis and throughout my studies.

Eline FolkersGroningen, March 2021

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Glossary

CM Corrective Maintenance

CVRP Capacitated Vehicle Routing Problem

FCDP Fleet Composition and Deployment Problem

FCP Fleet Composition Problem

MILP Mixed Integer Linear Program

NAM Nederlandse Aardolie Maatschappij

OSV Offshore Support Vessel

OWF Offshore Wind Farm

PM Preventive Maintenance

POB Personnel On Board

SSV Safety Standby Vessel

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Contents

1 Introduction 6

2 Problem description 82.1 A brief overview of the FCDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Maintenance types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Transport types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Difficulties of the FCDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Literature review 133.1 Sequentially solving interrelated subproblems . . . . . . . . . . . . . . . . . . . . . . 133.2 Sequentially solving FCDPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Simultaneously solving interrelated subproblems . . . . . . . . . . . . . . . . . . . . 143.4 Research gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Methodology 174.1 Dividing the FCDP into subproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Input for the subproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3 Subproblem 1: Fleet Composition Problem . . . . . . . . . . . . . . . . . . . . . . . 184.4 Subproblem 2: Vehicle Routing Problem . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4.1 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.5 Subproblem 3: Assignment of corrective maintenance tasks . . . . . . . . . . . . . . 28

5 Computational study 305.1 Description of input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2.1 Homogeneous CM buffer choices . . . . . . . . . . . . . . . . . . . . . . . . . 325.2.2 Heterogeneous CM buffer choices . . . . . . . . . . . . . . . . . . . . . . . . . 345.2.3 Weather impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2.4 Effect of maintenance demand . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 Conclusion 40

7 Discussion 417.1 Model development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.2 NAM maintenance operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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1 Introduction

Natural gas accounts for almost a quarter of the global energy production, and its use as energysource is expected to stay high over the next decades. The offshore production of natural gas ac-counts for an estimated 30% of global gas reserves (International Energy Agency, 2020). To keep theproduction going, it is essential to regularly maintain the offshore gas installations. Cost-efficientlymaintaining these installations is an increasingly important issue as gas prices have gone down, andmaintenance costs are high.

One of the largest cost drivers in offshore maintenance are the transport types used for performingmaintenance (Halvorsen-Weare et al., 2017). The maintenance operations are typically executed byhelicopters or vessels on long-term contracts. The work that they support can be divided into twomain types of maintenance tasks: preventive maintenance (PM) and corrective maintenance (CM).In contrast to PM tasks, CM tasks cannot be scheduled in advance since they rely on the uncertainnumber of failures occurring at gas installations. The unexpected CM tasks often need to be exe-cuted as soon as possible. Hence, it is necessary to have transport types available that can quicklyperform these typically short maintenance tasks. Therefore, when deciding on a fleet compositionof vessels and helicopters, it is important to select transport types that are cost-minimizing onthe long-term and have enough capacity to perform both types of maintenance tasks. This studymainly focuses on the strategic and tactical decisions, in which we determine what transport typesto use for what type of maintenance operations, and how many days to deploy each type such thatall maintenance tasks can be completed.

There a several reasons why selecting an optimal fleet composition is difficult. First, the underlyingoperational decisions must be considered as well. Optimal strategic and tactical decisions dependon decisions that are made at the operational level. Namely, the number of personnel scheduled andthe routing of vessels determine how many maintenance hours a transport type can perform on oneday, and hence affect the daily capacity of a transport type. It is necessary to make assumptionsabout operational decisions when deciding on a tactical fleet composition and deployment, becausethe operational planning of maintenance operations is already shown to be computationally chal-lenging (Irawan et al., 2017; Schrotenboer et al., 2018). Second, the transport types that can beused for maintenance operate in different ways. Some types can stay offshore for only one day andother types can stay offshore for several weeks or even months. In addition, the cost components ofthe transport types in the fleet can be rather specific and often depend on what types of tasks theyare used for. To represent reality as much as possible, the transport types must be modeled accord-ingly. Third, weather conditions affect the availability of the transport types considered: when windspeed or wave height is too high, the vessel or helicopter may not be deployed. Lastly, the randomevents consisting of possible failures of gas installations requiring CM tasks add stochasticity to theproblem. Both the uncertainty in weather and maintenance demand make it even harder to solvesuch a large size maintenance planning problem.

A lot of research on maintenance planning problems and Fleet Composition and Deployment Prob-lems (FCDPs) has been done, especially in the field of offshore wind farms. However, most of thedeveloped models cannot be applied directly to the problem addressed in this thesis. Namely, weconsider a large size offshore FCDP faced by the Nederlandse Aardolie Maatschappij (NAM), agas exploration and production company in the Netherlands. The transport types considered by

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the NAM require specific modeling and existing models cannot be directly implemented for thisreal-size problem.

This thesis attaches great importance to developing a method that can be used in practice. Wepresent a new approach for solving FCDPs at offshore gas installations. We divide the probleminto three solvable deterministic subproblems and develop a model for each subproblem. The firstmodel determines the cost-minimizing fleet composition to be used for both PM and CM tasks.The second model determines the routing of vessels and helicopters for the planned PM tasks, giventhe cost-minimizing fleet composition from the first step. We model this second subproblem as thewell-known Capacitated Vehicle Routing Problem (CVRP). Our third model is a simple insertionheuristic: we assign unexpected CM tasks to the fleet after simulating a Poisson process for eachgas installation individually. The idea of this methodology is that we predict the optimal strategicand tactical solution for this stochastic problem and evaluate it on an operational level afterwardsby observing the fraction of completed tasks. Our models are successfully implemented and testedon a large size FCDP faced by the gas exploration and production company NAM. Results of themodels show that climate transitions and a change in maintenance demand have a large effect onthe optimal fleet composition and corresponding cost. Our models provide decision makers insightsinto the cost and performance of fleet compositions under different parameter settings, enablingthem to make strategic decisions that minimize maintenance costs on the long-term.

The remainder of this thesis is structured in the following way. In Section 2, we describe the FleetComposition and Deployment Problem (FCDP) for maintenance at offshore gas installations. InSection 3, we examine the relevant literature on FCDPs and similarly structured problems. Ourmethodology is presented in Section 4. Here, we describe how the problem is divided into threesubproblems and we present the models developed for each of them. Next, the implemented modelsare tested on a real FCDP problem faced by the NAM. The results of that computational study arepresented in Section 5. Based on the results, a conclusion is given in Section 6. We conclude ourwork by discussing the limitations of our methods and present recommendations for future work inSection 7.

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2 Problem description

In this section, the Fleet Composition and Deployment Problem (FCDP) for maintenance at offshoregas installations is described. We start by giving a brief overview of the FCDP’s main concepts.Thereafter, the concepts that define this problem are introduced. First, the maintenance typesoccurring at offshore gas installations will be addressed. Second, the transport types that canperform this offshore maintenance are discussed. The differences between the transport types willbe described, and the cost components for offshore maintenance will be introduced.

2.1 A brief overview of the FCDP

The FCDP includes several candidate vessels and helicopters that can be used to perform main-tenance operations at offshore gas installations. During these maintenance operations, preventiveand corrective maintenance tasks are performed. The vessels and helicopters used to supportmaintenance activities are one of the largest cost drivers in the operational phase of offshore gasinstallations (Halvorsen-Weare et al., 2017). Hence, it is important to select a cost-effective fleetcomposition of vessels and helicopters that significantly reduce maintenance costs. It is essentialto take this decision for the long-term as transport types used for maintenance are contracted forseveral years. Therefore, the objective is to determine a fleet composition that minimizes long-termmaintenance costs, such that all maintenance tasks can be performed within the planning horizon.We quantify the maintenance tasks in the duration they take. The transport types that can carryout maintenance provide a certain capacity measured in hours. The total maintenance costs aredetermined by charter, personnel and travel costs of the fleet’s vessels and helicopters.

In the FCDP, we take decisions on three levels. First, strategic (long-term) decisions are made thatdetermine the general policy and broadly shape the operating strategies in the system (Crainic,2000). Once the strategic decisions are made, more practical issues can be decided on in the tac-tical (medium term) level. Finally, the operational (short-term) level concerns decisions related to‘day-to-day’ operations. More specifically, for the FCDP, we must strategically decide what trans-port types to use for both preventive and corrective maintenance operations. Then, at the tacticallevel, a decision is made on how days to deploy each type such that all maintenance tasks can becompleted. Hence, decisions in the first two levels determine the fleet composition. Lastly, at theoperational level, we determine the deployment of the fleet.

We mainly focus on the strategic and tactical level, and the operational level mainly serves as afeasibility check of the decisions made in the first two levels. During all three levels, we must takeinto account weather uncertainties as well as uncertainty in the number of failures that lead tocorrective maintenance operations.

2.2 Maintenance types

We consider two types of maintenance tasks that can be performed during a maintenance operation:preventive and corrective maintenance tasks. Preventive maintenance (PM) tasks are carried out toprevent failures. Corrective maintenance (CM) tasks are required when unforeseen events occur orwhen safety must be restored at a gas installation. We assume the number of planned maintenancehours is given at the beginning of the planning horizon and they can thus be scheduled in advance.Corrective tasks on the other hand, cannot be planned, since they rely on the uncertain number

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of failures. It is important to execute corrective maintenance tasks as soon as they occur, becausedowntime costs are incurred for each postponed corrective task. Therefore, capacity of transporttypes will be reserved for the corrective maintenance hours. More details on how this is done willbe explained later in Section 4.

The uncertain number of corrective maintenance tasks affect solutions to the FCDP since thechosen fleet composition should be able to quickly perform them next to the planned preventivetasks. Solutions are feasible only if a transport type can be made available soon after an unexpectedcorrective maintenance task is enforced. One can choose, for example, to charter a helicopter for aCM task that transports only a few people to the gas installation demanding this task. The otheroption is to interrupt a vessel’s preventive maintenance operation, and decide to let this vessel makea detour from its scheduled route. For a small task, chances are that a lot of personnel on board ofthe vessel cannot be used, even though day have to be paid. Both options lead to extra costs, anda trade-off must be made between the different options every time a corrective maintenance task isdemanded.

2.3 Transport types

We consider three transport types that are used for preventive and corrective maintenance oper-ations: barges, Offshore Support Vessels (OSVs) and helicopters. Barges are large mobile, self-elevating work platforms, with several movable legs attached to it [30]. Once the legs are placed onthe sea bed, the platform can lift itself up with a jacking system, as depicted in Figure 1a. Becauseof the crane(s) and large accommodated facilities, a barge is ideal for the heavier and more timeconsuming maintenance operations [12]. For this reason, barges are only deployed for preventivemaintenance operations. Barges stay offshore for several years and do not have to return to aharbor after each maintenance operation. Towboats can move the barge from one gas installationto another, which takes several days. Equipment and personnel are transported to the barge byhelicopters or vessels. A barge has room for at most maintenance personnel, where half of thepersonnel work day shifts and the other half work night shifts. The barge is the only transport typewe consider to work day and night, the other transport types only work at daylight.

The second type of transport that can be used is an OSV. This vessel is equipped with a motioncompensated gangway that allows personnel to be easily and safely transferred aboard an offshore

Table 1: Characteristics of transport types in the fleet composition and deployment problem(FCDP).

Transport Max. capacity Maximum Maximum operational threshold Return to Dailytype for personnel speed harbor productivity

(km/h) Wind speed (m/s) Wave height (m) (hrs/day)

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gas installations without a helicopter deck. This means that helicopters can perform maintenanceon gas installations. Unlike the vessels and barges, helicopters do not stay at aninstallation during a maintenance operation. It transports personnel to and from the installation atthe beginning and the end of the day. Because of this transport time, personnel operating from ahelicopter have a lower daily productivity than personnel operating from an OSV or barge. Anotherdisadvantage of using helicopters, is that personnel can only bring a limited number of tools, andno crane is at their disposal. Hence, helicopters are suitable for the lighter and less-time consumingmaintenance operations. Because of their high speed, helicopters can move personnel to installa-tions much quicker than the other types, which is beneficial for corrective maintenance operations.

One last transport type characteristic that needs to be discussed is the operational threshold. Wheneither the wind speed or wave height is above a certain mode-dependent threshold (as listed in Table1), the mode of transport cannot perform maintenance. In those cases, we will set the capacity ofa transport type to zero.

Next, we discuss maintenance costs associated with the three transport types. A daily fixed cost isassigned to each type. These costs depend on the daily charter cost and on the number of personnelon board. The fixed cost for an OSV and barge must be paid for all scheduled days in the shift,even if the mode of transport cannot operate one or more days due to bad weather conditions forexample, because these vessels do stay out at sea. This is different from the helicopter, which staysin the harbor during bad weather. In addition to the fixed cost, two types of flexible costs areincurred when a barge is used for maintenance operations. First, some gas installations requireextra safety precautions for the large number of personnel that operates from a barge. When abarge operates at those gas installations, a more expensive type of Safety Standby Vessel (SSV) isused. Hence, daily costs are increased when a barge is used there. This additional SSV is not neededwhen helicopters or OSVs operate at those installations. The reason is that OSVs are equippedwith rescue boats, offering enough space for everyone on board. For personnel operating fromhelicopters, rescue vessels that are already at sea can serve as safety vessel. Secondly, a high cost isincurred every time a barge moves to a new installation. Hence, the use of a barge is presumptivelyefficient only if it can perform maintenance a substantial number of days in a row at the sameinstallation.

2.4 Difficulties of the FCDP

We have seen that each transport type has its own characteristics. Some characteristics make atransport type less attractive, although its cost might be low. The transport type with the lowestcost is not always the optimal choice, because in general, this type has more limitations than theother types in the fleet. For example, a helicopter has low cost when it travels to an installation thatis close to the harbor. However, a helicopter has low capacity and low daily productivity. To arriveat an optimal fleet composition, these kinds of trade-offs must be made for every gas installationand maintenance demand.

The second and main reason why the FCDP is so complex, comes from the fact that operationaldecisions are intertwined with the strategic and tactical oriented FCDP, as mentioned in Section2.1. To be able to make strategic and tactical decisions, we need to make assumptions about theoperational decisions. After observing the optimal fleet composition, the deployment (operational)

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part of the FCDP must be solved. Only then we are able to check whether the strategic and tacticaldecisions are feasible, and if our assumptions were reasonable.

Lastly, two types of uncertainty must be taken into account: weather uncertainty and the number ofcorrective maintenance tasks. The latter is already discussed in Section 2.2. Weather conditions playan important role in the FCDP, because a transport type may not execute maintenance operationswhen either the wind speed or wave height is above its given operational threshold. The weatherconditions will determine the daily capacity of each transport type. The capacity is set to zerowhen the average wind speed or wave height on that day is too high. More details on how weatheruncertainty is taken into the model will be discussed later in Section 4. One last thing to note isthat the vessels need to be booked in advance, and the corresponding costs are incurred even if thevessel cannot perform maintenance due to bad weather circumstances. Hence, weather conditionsinfluence the optimality and feasibility of solutions to the FCDP.

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3 Literature review

In the problem description, we explained that decisions in the FCDP take place on three levels.Such a problem can be divided into smaller interrelated subproblems. In Section 3.1, we willaddress literature that tackles planning problems by means of sequentially solving interrelatedsubproblems. Also in the field of FCDPs, such an approach is widely used, to be reviewed inSection 3.2. Then, in Section 3.3 we elaborate on more recent literature in which methods aredeveloped that simultaneously solve the subproblems. We conclude this section by discussing theadvantages and disadvantages of both and we address the research gap in Section 3.4.

3.1 Sequentially solving interrelated subproblems

Not only in the field of FCPs, but in various applications, tactical and operational planning problemsare interrelated. Before diving into literature about FCPs specifically, let us briefly discuss papersthat have investigated interrelated subproblems as well. In the context of freight transportation,Balakrishnan et al. (1987) discuss a modeling approach for the underlying Location Routing Prob-lem (LRP): a combination of the Facility Location Problem (FLP) and the Vehicle Routing Problem(VRP). They sequentially solve the tactical and operational oriented problems. First, the solutionto the FLP is found by minimizing the sum of depot-to-customer distances. Thereafter, the routingproblem is solved based on the location of the depots from the solution obtained in the first problem.

In resource capacity planning and healthcare systems, planning decisions are often strongly inter-related (Hulshof et al., 2012). An example for the former is that capacity dimensioning of beds,equipment and staff for one care unit can affect the operations of other care units (Smith-Daniels etal. (1988); Akkerman and Knip (2004); Cochran and Roche (2009); Bretthauer et al. (2011)). Anexample for the latter are Operating Room (OR) planning problems concerning both the allocationof OR block times to surgical sub-specialties and the assignment of patients to wait-lists, wheresurgery durations are uncertain (Testi and Tanfani (2009); Denton et al. (2010)).

Larsen et al. (2019) consider an application in context of load planning, where containers needto be assigned to rail-cars. The tactical capacity planning solution depends on the solution of apacking problem at the operational level. They developed a two-stage stochastic approach to predictthe tactical solution under imperfect information by sampling and solving operational probleminstances.

3.2 Sequentially solving FCDPs

Maisiuk and Gribkovskaia (2014) address the planning problem of supply vessels to service off-shore oil and gas installations, where vessels are chartered on long-term contracts. They present adiscrete-event simulation model that can be used to determine the cost-efficient fleet compositionand size for annual supply operations. Halvorsen-Weare and Fagerholt (2017) consider an OffshoreSupport Vessel (OSV) planning problem for offshore oil and gas installations as well. The goal oftheir problem is to decide on an optimal fleet of OSVs as well as to determine their optimal routingand scheduling for maintenance operations. They present a new arc-flow and voyage-based modelto solve both decision problems.

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Other offshore planning problems arise in the wind energy industry. Halvorsen-Weare et al. (2013)consider the problem of determining the optimal vessel fleet to support maintenance operations atOffshore Wind Farms (OWFs). They developed a deterministic optimization model for this vesselfleet composition problem. However, real world FCPs are highly affected by uncertainty in weatherand failures that lead to corrective maintenance operations. In some situations, a deterministicapproach can still generate insights by solving the problem several times for different realizationsof the uncertain parameters. However, in order to obtain a robust and (nearly) optimal solution,other ways of incorporating uncertainty need to be investigated.

Halvorsen-Weare et al. (2013) use an optimization model to determine optimal vessel fleets forOWFs. According to the literature review by Hofmann (2011), a majority of the models use sim-ulation tools to analyze maintenance costs. The following papers have studied the FCDP with themain focus on tactical decisions as well, and use simulation techniques to either estimate or evaluateoptimal fleet compositions. Van de Pieterman et al. (2011) use an Operation and MaintenanceCost Estimator (OMCE), developed by Rademakers et al. (2008), to estimate the optimal fleetsize for maintenance at an hypothetical wind farm located off the coast of the Netherlands. Vande Pieterman et al. (2011) use a time-domain simulation program, whereas Dalgic et al. (2015)use Monte-Carlo simulation. Instead of estimating an optimal fleet, they evaluate different fleetsand select the fleet that minimizes total maintenance costs thereafter. Vis et al. (2005) developedan Integer Linear Programming (ILP) model to determine the vehicle fleet size for transportationof containers at a terminal. They use simulation to validate the estimations obtained from theiranalytical model. Calvete et al. (2007) apply Mixed Integer Programming (MIP) in combinationwith a two-step approach called an ’Enumeration-Followed-By-Optimization’ (EFBO) approach.

When determining the optimal fleet composition for maintenance operations, it is important todecide on the routing of vessels and scheduling of resources on board of the vessels as well (Hoffet al., 2010; Redmer et al., 2012). The following papers investigate FCDPs while focusing moreon the operational decisions. Dai et al. (2015) were the first to present a Mixed Integer LinearProgramming (MILP) formulation for the routing and scheduling problem of a maintenance fleetfor OWFs. Stalhane et al. (2015) present a similar model and decompose it by the Dantzig–Wolfedecomposition method. The re-formulated problem is decomposed into a master problem in whichvessels are assigned to maintenance tasks, and a subproblem that involves assigning routes andschedules for each vessel. Irawan et al. (2017) extend their research by proposing a new mathemat-ical model that takes into account multiple O&M bases and wind farms. They go even further byconsidering different skilled technicians, spare parts and ability of a vessel to transfer spare partsat each O&M base.

3.3 Simultaneously solving interrelated subproblems

More recently, papers have been published that deal with simultaneously solving interrelated sub-problems. Rademeyer and Lubinsky (2017) created a decision support system for on-the-roadpersonnel scheduling, where all three decision levels are dealt with simultaneously. In their opti-mization approach, all tactical decisions, for example, are evaluated right down to the operationallevel to check if the solution is feasible.

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Gundegjerde et al. (2015) were the first to consider stochastic programming approaches for solvingthe fleet size problem in the offshore wind industry. They propose a three-stage stochastic pro-gramming model in which they consider uncertainty in vessel spot rates, weather, electricity andfailures to the system. In the first and second stage, decisions on the fleet composition are made.In the third stage, deployment decisions of that fleet composition are considered. For only a limitednumber of scenarios, their stochastic model was able to find the optimal combination of decisions.A few years later, Stalhane et al. (2019) proposed a two-stage stochastic programming model. Intheir model, the first stage decisions are what vessels to charter. The second stage decisions are todetermine which maintenance tasks to support by what vessels given the vessel fleet compositionfrom the first stage. They generated multiple scenarios for the uncertain weather conditions andfailure occurrences. The two-stage model is solved given a specific scenario’s realization of the un-certain parameters. In this way, the authors solved the extensive form of the deterministic problem.

Schrotenboer et al. (2018) focused on the operational level of maintenance at OWFs by jointlyoptimizing the sharing of technicians between wind farms and routing of the technicians that needto be delivered and picked up. They simultaneously determine the optimal allocation and routingproblem by using an Adaptive Large Neighborhood Search heuristic that finds computational effi-cient and high quality solutions.

Gutierrez-Alcoba et al. (2019) provide a Mixed Integer Linear Programming (MILP) of a modelthat optimizes the vessel fleet composition for an OWF. This model simultaneously decides, on thehigher level, the optimal fleet composition and on the lower level the scheduling of maintenanceoperations for a set of weather and breakdown scenarios. As a result, the maintenance cost corre-sponding to the optimal fleet composition is an underestimation compared to what can be obtainedunder incomplete information. Therefore, they present a heuristic that simulates scheduling underimperfect information for a better cost estimate.

Schrotenboer et al. (2020) examined the tactical decision making in offshore wind maintenanceservice logistics. They provide a two-stage stochastic MIP model, where vessels are assigned todepots in the first stage, and to maintenance tasks in the second stage. Maintenance tasks andweather conditions are revealed in between the two stages. In contrast to many other studies onOFWs, they take the viewpoint of one maintenance service provider responsible for multiple windfarms.

The recent work of Premkumar and Kumar (2020) in the field of railway networks is one of the fewpublished papers in which the strategic, tactical and operational decision levels are combined intoone model. They investigate the locomotive assignment problem where the objective is to assigna fleet of locomotives at minimum cost to pre-scheduled trains. They proposed a robust linearmathematical model for solving the problem, in which they integrated the three decisions levelssimultaneously. A Dijkstra’s algorithm-based heuristic is developed by them to solve the integratedproblem. Their model was able to converge to an optimum only in case of small problem instances.

Simultaneously solving interrelated subproblems offers the promise of finding more optimal solutions(Balakrishnan et al., 1987). The advantage of integrating multiple decisions at once is that ”we donot find ourselves battling at the operational decision level with infeasible decisions that were madehigher in the hierarchy” (Rademeyer et al., 2017). However, incorporating all decisions together

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in one model becomes computationally impractical for larger problem instances (Wu et al., 2002).Although sequentially solving subproblems often leads to sub-optimal solutions, solving the smallersubproblems one by one allows us to solve the FCDP for larger (real-world size) problem instances(Larsen et al., 2019).

3.4 Research gaps

Based on the literature review conducted above, we can conclude that a lot of research has beendone on maintenance at offshore wind farms. However, the operations and maintenance issues atoffshore gas installations are not the same as in the offshore wind energy sector. In the first place,maintenance operations at offshore gas installations occur less frequently, and often, the durationof one operation is much longer compared to operations on wind turbines. In the second place, thedispersed nature of wind turbines leads to the fact that 2-5 technicians may be required on multipleturbines, as opposed to around 15-30 technicians on one large gas installation (Baldock et al., 2014;Jonker et al., 2017). As a result, different types of transport are demanded for maintenance at gasinstallations.

Several papers have been devoted to optimizing strategic decisions for maintenance logistics withinOWFs, while considering the operational decisions as well. Many papers focus on developing modelsfor a general setting of wind farms, but often those models are not directly applicable to real-sizewind farms. A pitfall of modeling a general wind farm setting is that more decisions than needed areadded into one model to fit as many real-world cases as possible. We have seen that simultaneouslytaking decisions at multiple levels lead to more complex models and is often at the expense of thesolvability of larger problem instances. This thesis, on the other hand, attaches great importance todeveloping a method that can be used in practice. Our proposed models only contain the relevantdecisions, resulting in models that are solvable for real-size problems. Moreover, we can give moreguarantee that the resulting solutions can be implemented in practice.

In this thesis, we present a new pragmatic methodology for integrally scheduling maintenance op-erations that can be used for real size offshore fleet composition and deployment problems. Wedivide the problem into three solvable deterministic subproblems, where tactical decisions are madein the first subproblem, and are evaluated in the second and third subproblem thereafter. In allsubproblems, we deal with the uncertainty of weather and failures leading to corrective mainte-nance (CM) tasks. We do this by incorporating a buffer into our model, rather than incorporatingthat uncertainty directly in a stochastic optimization approach. As a result, uncertainty is takencare of indirectly and the approach remains tractable. Using this procedure, we are able to finda solution to the fleet composition and deployment problem that can be assessed for a large sizeof CM scenarios. The proposed models can provide decision makers insights into the cost andperformance under different scenarios of weather and maintenance demand, but also into the effectof operational decisions such as crew-sizing and allocation of helicopter decks.

16

4 Methodology

In this section, we discuss the methodology that was used to solve the FCDP. First, we describehow the master problem is divided into smaller subproblems in Section 4.1. In addition, we explainhow these subproblems are related to each other. Section 4.2 elaborates on the input that is neededfor solving the subproblems. Then, in Sections 4.3, 4.4 and 4.5, we discuss each subproblem in moredetail.

4.1 Dividing the FCDP into subproblems

As discussed in Section 2, the FCDP integrates three types of decisions, which cannot be opti-mized jointly without resulting into computational unsolvable models. We, therefore, propose analternative three-step approach to integrally schedule offshore maintenance operations. We dividethe FCDP in three subproblems that are solved in sequence, where the output of the first problemdetermines the input of the second problem, and the the output of the second problem determinesthe input of the third problem. See Figure 2 for a graphical overview. Although solving the in-terrelated subproblems one by one can lead to sub-optimization, it allows us to solve the FCDPfor larger (real-world size) problem instances [27]. To deal with the uncertainty in our problem,we incorporate a buffer into our model, rather than incorporating that uncertainty directly in astochastic optimization approach. In this way, uncertainty is taken care of indirectly and the ap-proach remains tractable. This allows us to solve the stochastic problem in a deterministic setting.That is a big advantage, since often stochastic problems are much harder to solve than deterministicproblems [14]. In Section 4.2 we will describe which input is needed to solve the subproblems. InSections 4.3, 4.4 and 4.5, we will clarify how the three subproblems are connected, and explainwhat methods are used to solve each subproblem.

4.2 Input for the subproblems

The first input needed to solve the FCDP is the set of transport types that is available for off-shore maintenance. We determine the number of barges, OSVs and helicopters that can be usedthroughout the entire planning horizon. The second input for our model is the given number ofpreventive maintenance (PM) hours per gas installation that needs to be performed within theplanning horizon.

Figure 2: Graphical representation of the methodology.

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In the first subproblem, we reduce the capacity of each transport type to leave sufficient time fortransport and corrective maintenance (CM), which is to be determined in the second and thirdsubproblem. We reserve capacity per transport type, so that we can flexibly choose the availabil-ity for CM of each type. The three subproblems will be solved for different buffer choices. Aftershowing the results of all buffer choices, it is up to the decision-maker to determine which types oftransport he wants to deploy for PM and CM tasks. Hence, the third input of the model is a bufferfor transport and CM for each transport type separately.

As indicated in Figure 2, the last input for the three models is a weather scenario. We generated aweather scenario, based on historical data, that predicts future weather outcomes. We assume thatfuture weather behaves more or less the same as the past. As a result, we will get output that isbased on average weather circumstances.

The following procedure was used to obtain a relevant and reliable weather scenario. In our problemsetting, the operational conditions of transport types are affected only by wind speed and waveheight. Therefore, we collected historical data on these two weather conditions for the installationlocated at the center of the gas field (Rijkswaterstaat, Platform K13a, 2011-2019). Because windspeed and wave height were respectively given in time steps of 60 and 10 minutes, we calculated theaverages over the day. For this strategic problem, taking only one location with daily observationssuffices. We followed the weather simulation approach presented by Halvorsen-Weare & Fagerholt(2017). Because weather of consecutive days is dependent, we used a Markov chain process forgenerating weather outcomes. We classified for both wind speed and wave height three weatherstates, based on the operational thresholds of the transport types given in Table 2. For each dayin the dataset, we translated the daily averaged wave height to the corresponding states. As aresult, we have created a sequence of wave states, called a Markov chain. To incorporate seasonaleffects, we created transition probability matrices for each season separately based on the slices ofthe Markov chain corresponding to that season. The transition matrices calculate the probabilityof moving from one wave state to another. Based on the transition matrices of all four seasons, wegenerated a wave state scenario for one entire year. To preserve the strongly correlated wind states,we created a wind state scenario from the generated wave state scenario given their correlation.

Table 2: Wind states and wave states.

Wind Wind speed Wave Wave heightstate (m/s) state (m)

1 12 23 3

4.3 Subproblem 1: Fleet Composition Problem

In the first subproblem, we both decide at which periods each available transport type is deployedand decide at which periods PM of each platform is scheduled. We do not decide what specificmaintenance tasks will be performed by what (types of) vessels and helicopters. Rather, we ensurethat for each period, enough vessels and helicopters are deployed to cover the PM being scheduled.

18

A deterministic Mixed Integer Linear Program (MILP) is used to find the optimal fleet composition.This MILP formulation is based on the first stage model framework presented by Stalhane et al.(2019) that determines what vessels to charter to support maintenance at an offshore wind farm.The formulation is adapted for NAM specific use, as we will indicate later. We start by introducingall relevant sets and parameters of the MILP before the formulation is presented. An overview ofthe notation is given in Table 3.

Let G be the set of all offshore gas installations in the field. The subset Gh ⊂ G contains all instal-lations with a helicopter deck. In addition, we consider a subset of gas installations G′ ⊂ G thatrequire additional maintenance costs due to extra safety precautions. Namely, when a barge op-erates at an installation g ∈ G′, a more expensive type of Safety Standby Vessel (SSV) must be used.

Each installation g ∈ G requires a certain number of preventive maintenance man-hours Dg. Let Tbe the set of days in the planning horizon. All PM hours must be performed within the length |T |of the planning horizon.

Maintenance can be executed by different modes of transport. Let V be the set of all transporttypes that can be used. In addition, we consider subsets of V, because some constraints in themodel will be defined for only part of the transport types v ∈ V. The set V ′ ⊂ V consists of alltransport types excluding the barge. The OSVs are exclusively included in VW ⊂ V. Finally, theset VH ⊂ V solely includes the helicopters.

The total cost per day of using a transport type v ∈ V ′ is denoted by Cv. This includes daily charterand personnel cost. The OSVs are contracted for the entire planning horizon. This means that thevessels are available for maintenance on all days in the planning horizon. We add a penalty cost Cp

v

for each day that a vessel v ∈ Vw is not deployed. The total cost of using a barge for maintenancecomposes of three price components. First, a daily cost Cb is incurred. This includes daily charter,personnel and supply vessel costs. Second, an additional daily cost Cb

g is added when the bargeperforms maintenance on a gas installation g ∈ G′. Third, demobilizing and mobilization costs Cb,m

are charged for every time the barge moves to another installation.

The number of personnel operating from transport type v ∈ V is denoted by Nv. Let Hv be thenumber of effective hours one person can perform while traveling with transport type v on an av-erage working day. Then the maximum daily capacity of transport type v can be computed asCapv = Nv ·Hv. The capacities of the transport types are not the same everyday. Therefore, weintroduce the parameter Capvt denoting the maximum capacity of transport type v ∈ V on dayt ∈ T . There are two cases in which Capvt 6= Capv, to be considered in the next two paragraphs.

19

Table 3: Overview of sets, parameters, and decision variables in the first subproblem.

Set definitions

T = {1, . . . , T} Set of days in the planning horizon.G Set of all gas installations.G′ ⊂ G Set of gas installations that require additional cost when a barge is used.Gh ⊂ G Set of gas installations with a helicopter deck.V Set of all transport types.V ′ ⊂ V Set of transport types excluding the barge.Vw ⊂ V Set of OSV vessels.Vh ⊂ V Set of helicopters.

SFg

i Set of periods in the planning horizon with equal length, i ∈ {1, 2, . . . , 2 · Fg}.P

Fg

i Set of periods in the planning horizon with equal length, i ∈ {1, 2, . . . , Fg}.

Parameters

Cv Total cost per day to use transport type v ∈ V ′.Cp

v Penalty cost per day for not using a vessel v ∈ Vw that is contracted for the entire planning horizon.Cb Total cost per day to use the barge.Cb

g Additional cost of using a barge at installations g ∈ G′.Cb,m Cost of demobilizing and mobilizing the barge to move to a new installation.

Dg Planned maintenance hours required on installation g.Fg Yearly visit frequency of installation g.Ng Maximum number of personnel allowed on installation g.

Nv Number of personnel operating from transport type v ∈ V.Hv Number of hours one person can perform while traveling with transport type v ∈ V on an average working day.OWind

v Operational limit of transport type v ∈ V in terms of wind speed.OWave

v Operational limit of transport type v ∈ V in terms of wave height.Windt Average wind speed on day t.Wavet Average wave height on day t.

Capvt Number of maintenance hours transport type v ∈ V can perform on day t.Capvgt Number of maintenance hours transport type v ∈ V ′ can perform at installation g on day t.Capbgt Number of maintenance hours transport the barge can perform at installation g on day t.Mgt Maximum number of maintenance hours that can be performed at installation g on day t.

Decision variables

yvt Equals 1 if transport type v ∈ V ′ is used on day t.bgt Equals 1 if a barge is used at installation g on day t.xgt Number of maintenance hours performed at installation g on day t.dT Number of times the barge is moved during the planning horizon T .

Supporting variables

Xgt Equals 1 if installation g is visited on day t.bt Equals 1 if a barge is used on day t.zt Equals 1 if a barge is moved to a new installation on day t.

Let OWindv and OWave

v be the operational limits of transport type v in terms of wind speed andwave height, respectively. The average wind speed and wave height on day t are given by Windtand Wavet. We set

Capvt = 0, for v ∈ V and t ∈ T , when Windt > OWindv or Wavet > OWave

v .

It holds for the NAM specifically, that personnel operating from OSV vessels v ∈ Vw work in shiftsof l days. All vessels v ∈ Vw have a crew change in the harbor on a fixed day. During that day,

20

there is no time to perform maintenance with these vessels. Assume the planning horizon starts ona crew change day. Then, we set

Capvt = 0, for v ∈ Vw and t = {1, 1 + l, 1 + 2l, . . .}.

Let Ng denote the maximum number of people allowed on installation g. When the number ofpeople Nv on board of transport type v is higher than Ng, daily capacity should be set lower for thatinstallation specifically, because part of the personnel cannot be used. In that case, the maximumcapacity that transport type v ∈ V ′ can use at installation g on day t equals min(1,

Ng

Nv) · Capvt.

This expression ensures that when Ng < Nv, the daily capacity at g is reduced by a fractionNg

Nv,

and Capvgt = Capvt otherwise. However, for OSVs, it is possible to divide personnel over twoinstallations on one day, called dual manning. At the start of the day, half of the personnel (Nv/2)is dropped of at a second installation nearby. Dual manning is convenient when Nv > Ng on eitherlocation g, since more personnel can be deployed and less OSV capacity is wasted. Unfortunately,dual manning cannot always be realized, due to practical reasons (e.g. the crane on the OSV isneeded on both installations) or safety reasons (e.g. no safety vessel nearby the second installation).We assume that dual manning can be realized half of the time. Hence, on average, Nv/1.5 people arelocated at one installation at the same time. It follows that the maximum capacity that transporttype v ∈ Vw can use at installation g on day t is given by

Capvgt = min(

1,Ng

Nv/1.5

)· Capvt, v ∈ Vw. (1)

Dual manning with helicopters is not possible, thus for v ∈ Vh, it remains true that

Capvgt = min(

1,Ng

Nv

)· Capvt, v ∈ Vh. (2)

Given (1) and (2), we can compute the maximum capacity that can be used at installation g onday t by Mgt = max(v ∈ V ′ : Capvgt). For a barge however, the computation of the installationspecific capacity is a bit different. The first reason is that dual manning cannot be executed bypersonnel operating from a barge. Secondly, half of the barge personnel work day shifts and theother half work night shifts. Hence, a barge effectively has Nv/2 people working at the same time.In addition, daily capacity Capvt is multiplied by two, because a barge operates with two shifts eachday: a day and night shift. Hence, the maximum capacity that the barge can use at installation gon day t is computed as

Capbgt = 2 ·min(

1,Ng

Nv/2

)· Capvt, v ∈ V \ V ′.

Some gas installations need multiple visits during the planning horizon, because several mainte-nance tasks need to be carried out regularly. Therefore, we introduce visit frequencies Fg for eachinstallation g. This parameter indicates the minimum number of visits required at installation g

during the planning horizon. Let SFg

i be sets of periods in the planning horizon with equal length,where i ∈ {1, 2, . . . , 2 · Fg}. As an example, for installations with a visit frequency of two peryear, we define four sets: S2

1 = {1, . . . , 91}, S22 = {92, . . . , 182}, etc, assuming T = 365. During

all these periods, the installation requires at least one visit. In this example, creating two sets isnot enough: maintenance on the installation could than be performed on the last day of the first

21

period, and on the first day of the second period. In that case, the installation is visited once in-

stead of twice during the planning horizon. Therefore, we create 2·Fg sets SFg

i for all installations g.

Some maintenance tasks always require a minimum number of working hours. In addition, plannedmaintenance should be spread equally over the planning horizon. Therefore, an installation needs tohave a minimum visit duration during each visit. We define, for each installation g, sets of periods

PFg

i in the planning horizon with equal length, with i ∈ {1, . . . , Fg}. During each period PFg

i , partof the total demand Dg at installation g needs to be performed.

There are four decision variables. First, the binary variable yvt indicates whether transport typev ∈ V ′ is used on day t. Second, the binary variable bgt indicates whether a barge is used at installa-tion g on day t. For a barge, it is important to keep track of its location during the whole planninghorizon, because we have to count the number of times it moves to another installation. Only thencan we correctly compute the total cost of the barge in the model. Therefore, the second decisionvariable has an index g, in contrast to the first decision variable yvt. For the other transport typesv ∈ V ′, it is not important to know exactly where they are used, since the cost of helicopters andOSVs are independent of the installation they visit. (Thus, the location of the barge is fixed by thefirst model, the location of the vessels and helicopter is determined later in the second subproblem.)Still, we have to keep track of the number of maintenance hours that are completed at installationg on day t (by any of the types v ∈ V ′). For that, we introduce a third decision variable xgt. Thefourth decision variable dT denotes the number of times the barge is moved during the planninghorizon.

In addition, we consider three supporting binary variables. First, let

Xgt =

{1, if xgt > 0

0, otherwise

indicate whether installation g is visited on day t by any transport type v ∈ V ′. Second, the variablebt =

∑g∈G bgt equals one when the barge is used on day t. Third, zt is equal to one if the barge is

being moved on day t.

The objective is to determine a fleet composition that minimizes total maintenance costs, such thatall maintenance hours can be performed within the planning horizon.

Using the notation described above, the first subproblem can be formulated as follows:

min Z1 =∑t∈T

∑v∈V′

Cv · yvt +∑v∈Vw

Cpv ·(T −

∑t∈T

yvt

)(3)

+∑t∈T

(∑g∈G

Cb · bgt +∑g∈G′

Cbg · bgt

)+ Cb,m · dT , (4)

subject to∑g∈G

xgt ≤∑v∈V′

Capvt · yvt, ∀t ∈ T , (5)

22

xgt ≤∑

v∈V′\Vh

Capvt · yvt, ∀g ∈ G \ Gh,∀t ∈ T , (6)

xgt ≤Mgt, ∀g ∈ G,∀t ∈ T , (7)∑v∈V′

yvt =∑g∈G

Xgt, ∀t ∈ T , (8)

(l−1)− (l−1)(1− yvu) ≤l−1∑w=1

yvu+w ≤ (l−1) · yvu, ∀v ∈ Vw, (9)

∀u ∈ {1, 1 + l, 1 + 2l, . . .},

∑g∈G

bgt ≤ 1, ∀t ∈ T , (10)

bg,t − bg,t−1 ≤ zt, (bg,0 = 0) ∀g ∈ G,∀t ∈ T , (11)

dT =∑t∈T

zt, (12)

bgt ≤ 1−∑t

s=1 bst

+

∑ts=1 bgst

∀g ∈ G,∀t ∈ {1, . . . , 5}, (13)

bgt ≤ 1−∑t−1

s=t−5 bs

5+

∑t−1s=t−5 bgs

5∀g ∈ G,∀t ∈ {6, . . . , T}, (14)∑

t∈SFgi

xgt + bgt ≥ 1, ∀g ∈ G with Fg > 1, (15)

∑t∈PFg

i

xgt + Capbgt · bgt ≥Dg

Fg, ∀g ∈ G. (16)

The objective function (3) minimizes the overall maintenance cost. The first term in the objectivefunction is the total cost for using helicopters and OSVs deployed during the planning horizon.The last three terms refer to the barge costs, consisting of daily charter cost, additional cost foroperating on installations g ∈ G′, and moving cost, respectively.

Constraint (5) ensures that total maintenance hours performed on day t does not exceed total ca-pacity of the active transport types on day t. Constraint (6) limits capacity that can be used oninstallations without a helicopter deck. Constraint (7) ensures that maintenance hours performedat an installation does not exceed the maximum capacity that can be used on that installation.Only one transport type is allowed on an installation each day, established by constraint (8). TheOSVs used by the NAM always work in shifts of l days. Constraint (9) ensures that an OSV isused all l days in the shift, and avoids the possibility of deploying an OSV only for a few days. Ifan OSV does not leave the harbor on the starting day u, then the OSV is not used the next l − 1days either.

Constraints (10) - (14) are specifically constructed for the barge considered in the model. Constraint(10) ensures that the barge performs maintenance on at most one installation per day. Constraints(11) and (12) ensure that moving costs of the barge are computed correctly. In constraint (11), the

23

binary variable zt is set to one when the barge starts working at a new installation. Constraint (12)counts the number times the barge is moved during the planning horizon. For the barge that canbe used by the NAM, it takes five days to move to a new installation. Constraints (13) and (14)ensure that the barge cannot be used on an installation if it was performing maintenance on one ofthe other installations in the previous five days.

Constraint (15) makes sure that each installation is visited with the required frequency. During

every period SFg

i , the installation needs to be visited at least once. Constraint (16) ensures that

performed maintenance hours at g are equally distributed over the periods PFg

i in the planninghorizon. At the same time, constraint (16) ensures that maintenance demand for the entire planninghorizon is met for each installation.

4.4 Subproblem 2: Vehicle Routing Problem

The purpose of the second subproblem is to construct cost-minimizing routes for preventive main-tenance (PM) operations performed by the fleet that we obtain from the first subproblem. Becausefailures leading to corrective maintenance (CM) tasks are revealed at the last moment (in prac-tice, usually after it has already been decided which routes will be sailed), the optimal routes willbe constructed only for PM operations. These routes will act as input for the third subproblem,because there we want to assign CM tasks to transport types based on their availability and location.

In this second subproblem, we will determine routes for each period of length l separately, becausethe OSV vessels always operate l days in a row before returning back to the harbor. The cost-minimizing routes can be found by solving a Capacitated Vehicle Routing Problem (CVRP). Thisproblem is often treated in the context of servicing demand of customers with a fleet of capacitatedvehicles based at a single depot. The setting of our problem is the same: we want to service PM de-mand of gas installations with a fleet of capacitated vessels and helicopters based at the same harbor.

The following decision variables of the first subproblem will be used as input for the CVRP. Firstly,the binary decision variable bgt reveals whether the barge is performing maintenance at installationg on day t. Based on that, and on the capacity Capbgt of the barge at that installation, we knowhow many PM hours were performed by the barge on installation g. These PM hours do haveto be considered in the CVRP, because we already assigned part of the installation specific PMdemand to the barge. Secondly, the binary variables yvt tell us whether the remaining transporttypes v ∈ V ′ are used on day t ∈ T . Thirdly, the decision variables xgt inform us on the number ofmaintenance hours that was performed on day t by any of transport types v ∈ V ′. Hence, for eachperiod consisting of l consecutive days, we know the availability of each transport type, and thePM demand of each gas installation. Given that, we assign PM demand to transport types withknown capacity through the construction of cost-minimizing routes.

The mathematical formulation for this CVRP is based on the formulation presented by Kallehaugeet al. (2005). They consider a vehicle routing problem with time windows. Deploying time windowsin a routing problem forces vehicles to arrive at a location within a given interval. Although wedo not have to deal with time windows, we can use the same decision variable as they defined fortracking a vehicles arrival time at each location. By using such a variable, we can ensure that thetransport types return to the harbor before the end of a period. In other words, it allows use to

24

Table 4: Overview of sets, parameters, and decision variables in the second-stage problem.

Set definitions

P Set of periods in the time horizon.L Set of nodes in the graph.L′ = L \ {0} Set of nodes representing gas installations.Lh Set of gas installations with a helicopter deck.V = Vw ∪ Vh Set of available transport types.Vw Set of OSV vessels.Vh Set of helicopters.

Parameters

Cijv Travel cost of transport type v associated with arc (i, j) ∈ A.Cp

v Penalty cost per extra maintenance hour to be executed by transport type v.Cb

v Bonus for each day transport type v is deployed less than scheduled.Tijv Travel time of transport type v associated with arc (i, j) ∈ A.Sv Travel speed of transport type v.Nv Number of personnel operating from transport type v.Capv Maximum number of maintenance hours transport type v can perform each day.Capjv Maximum number of maintenance hours transport type v can perform at installation j on one day.Capv(p) Total number of maintenance hours transport type v can perform during period p.Qv(p) Number of days transport type v can be used during period p.Dj(p) Planned maintenance hours required on installation j during period p.Nj Maximum number of personnel allowed on installation j.

Decision variables

xijv Equals 1 if transport type v traverses arc (i, j).sjv Arrival time of transport type v at installation j.rv Number of additional maintenance hours that can be performed by transport type v.Lv Number of days it takes (travel time and maintenance time) to traverse the route assigned to transport type v.

construct routes where a vessels’ travel time and maintenance time add up to at most l days.

4.4.1 Mathematical model

In this section, we present the mathematical formulation for the Capacitated Vehicle Routing Prob-lem (CVRP). We start by introducing all relevant sets and parameters used to describe the problem.An overview of the notation is given in Table 4. After that, we provide the decision variables for-mulated for the second subproblem. We end this section by presenting the corresponding MILPformulation with a derivation of each constraint.

The VRP is defined on a directed graph (L,A), where L is the set of nodes, and A the set of arcs.The fleet of transport types available for maintenance is based at the harbor, represented by node0. All routes must start and end at the harbor. The remaining node set L′ = L \ {0} represent thegas installations that need to be maintained. The set of arcs A consists of all possible connectionsbetween the nodes.

Let P be the set of periods in the planning horizon. Each period has a length of l days. It followsfrom the solution to the first subproblem how many preventive maintenance hours Dj(p) need to beperformed on installation j ∈ L′ during period p ∈ P. Maintenance can be performed by transporttypes v ∈ V. Let Capv be the maximum number of hours transport type v can perform maintenanceeach day. Let Qv(p) be the number of days transport type v is scheduled during period p, whichis given in the output of the first subproblem. We can compute the total capacity Capv(p) of

25

transport type v during period p by Capv(p) = Qv(p) · Capv. Similarly as in (1), the maximumcapacity that vessels v ∈ Vw can use at installation j on one day is given by

Capjv = min(

1,Nj

Nv/1.5

)· Capv.

A travel distance dij is associated to each arc (i, j) ∈ A. Given the travel speed Sv of transporttype v ∈ V, travel time of v associated with arc (i, j) is given by Tijv = dij/Sv. Then the travelcost of v associated with arc (i, j) is calculated as

Cijv = Cv · Tijv,

where Cv is the daily charter cost of transport type v.

The model of the CVRP contains the following decision variables. The first decision variable is xijvand equals one if transport type v traverses arc (i, j). The second decision variable sjv representsthe arrival time of transport type v at installation j. Finally, let rv be the number of additionalmaintenance hours that can be performed by transport type v. Per extra hour rv, a high penaltycost Cp

v must be paid.

A standard CVRP aims at finding cost-minimizing routes, and does not take into account locationpreferences of vehicles or duration of routes in terms of time. In our problem, we want to assignlarger vessels to larger installations, so that less capacity is wasted by having to much personnel onboard. In addition, we would like to construct routes in such a way that as many time as possibleis left for CM tasks. To achieve this, we add a bonus Cb

v to the objective function for each day atransport type v is deployed less than scheduled. By adding this, the model automatically assignslarger vessels to larger installations, because the larger vessels can perform more maintenance inshorter time. It is important that the parameter Cb

v is of the same proportion as Cijv, becauseboth the cost-minimizing and time-optimizing aspects are of equal importance. A fourth decisionvariable Lv is added, denoting the number of days it takes (travel time and maintenance time) totraverse the route that is assigned to transport type v.

The objective is to find, for each period separately, the set of cost-minimizing routes for each trans-port type, such that all maintenance hours at the installations can be performed. The routes mustbe feasible with respect to capacity of the transport types, the availability of the transport typesand the installations can only be visited by one transport type per period. In addition, the routesmust be constructed in such a way that transport types can perform maintenance in the shortesttime possible.

The mathematical formulation of the CVRP is given by

Z2(p) =∑v∈Vw

∑(i,j)∈A

Cijv · xijv +∑v∈Vh

∑(i,j)∈A

Cijv ·Dj(p)

Capjv· xijv (17)

+∑v∈V

Cpv · rv +

∑v∈V

Cbv · (Qv − Lv)

26

subject to∑v∈V

∑i∈L

xijv = 1, ∀j ∈ L′ (18)∑i∈L′

Di(p) ·∑j∈L′

xijv ≤ Capv(p) + rv, ∀v ∈ V, (19)

∑j∈L′

x0jv = 1, ∀v ∈ Vw, (20)

∑i∈L

xikv =∑j∈L

xkjv, ∀k ∈ L′,∀v ∈ Vw, (21)

∑i∈L′

xi0v = 1, ∀v ∈ Vw, (22)

x0kv = xk0v, ∀k ∈ L′,∀v ∈ Vh, (23)∑i∈L′

∑j∈L′

xijv = 0, ∀v ∈ Vh, (24)

∑j∈L′\Lh

x0jv = 0, ∀v ∈ Vh, (25)

siv +Di(p)

Capiv+ Tijv −M1(1− xijv) ≤ sjv, ∀i ∈ L,∀j ∈ L′,∀v ∈ V,M1 large (26)

siv +Di(p)

Capiv+ Ti0v −M2(1− xi0v) ≤ l − 1 +

rvCapv

, ∀i ∈ L,∀v ∈ Vw,M2 large (27)∑j∈L′

(∑i∈L

xijv

)· Dj(p)

Capjv≤ Qv(p) +

rvCapv

, ∀v ∈ V. (28)

The objective function (17) minimizes the overall transportation cost. Because a helicopter travelsback to the harbor at the end of the day, it has to traverse an edge multiple times when it doesmaintenance on one installation for several days. Therefore, the cost of traveling to location j is

multiplied by the number of daysDj(p)Capjv

the helicopter v ∈ Vh performs maintenance at location j.

Constraint (18) ensures that each installation is visited by exactly one transport type. Constraint(19) states that the number of maintenance hours a transport type can perform during the route islimited by its capacity. Constraint (20) ensures that each vessel leaves the harbor. Constraint (21)conserves the flow of the vessels at each node and constraint (22) indicates that each vessel mustreturn to the harbor. The flow of helicopters are conserved by the constraint (23). Helicopterscan only fly from the harbor to one installation per day, so helicopters always traverse an edge inboth directions. In addition, it means that edges in between installations are never traversed byan helicopter, established by constraint (24). Constraint (25) ensures that helicopters do not flyto installations without a helicopter deck. Constraint (26) establishes the relationship between thedeparture time of an installation and the installation that is visited next. Constraint (27) makessure that the vessels return to the harbor to change crew one day before the end of the period.Finally, constraint (28) limits the number of days a transport type can work during one route byits availability in that period.

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4.5 Subproblem 3: Assignment of corrective maintenance tasks

The goal of the third subproblem is to assess whether the fleet composition and deployment, re-sulting from the first two subproblems, is able to perform the unpredictable corrective maintenance(CM) hours next to the planned preventive maintenance (PM) hours. In the first subproblem,we estimated the number of days in one year that each transport type must be deployed, suchthat together, they can fulfill all PM and CM hours. There, we reserved part of the transporttypes’ capacity for CM hours. As a result, transport types are deployed more days than neededfor PM operations alone. Consequently, there is room for unexpected CM operations in the routesthat were constructed for PM operations in the second subproblem. For example, when we con-sider a period of 13 days and a vessel is assigned to a route that takes 10 days, there are 3 daysleft for CM operations. Similarly, when a helicopter is available for 6 days during one period, butit was deployed for only 4 days, that helicopter can be used for CM operations the remaining 2 days.

In subproblem 3, we evaluate how many CM tasks can be executed when the transport types travelthe cost-minimizing routes as constructed in the second subproblem. This will be done by simulat-ing multiple CM scenarios, generated in the following way. For each installation g, we are given theexpected number of CM tasks, say Qg, during one planning horizon. However, the failures leadingto these tasks occur at random. To model the arrival of events that occur at random, we use aPoisson process. For this, we need to compute the expected inter-arrival time λg of CM tasks. Thissimply equals λg = T

Qg, where T is the length of the planning horizon. Then, the arrival time Tgi

of the ith CM task at installation g is distributed as Tgi ∼ Gamma(i, λ), for i = 1, 2, . . .. Hence,each CM scenario is generated by sampling Gamma distributed events for each gas installation.

The insertion of CM tasks into routes will be done in the following way. Each scenario reveals,for all periods in the planning horizon, the timing of CM tasks with their corresponding gas in-stallations. From the second subproblem, we know for each transport type how many days arereserved for CM tasks, and we are informed about their location on the day a CM task is an-nounced. Then, a CM task is inserted into the route of an available transport type that is closestto the gas installation requiring corrective maintenance. By selecting the closest available type,we prevent vessels from taking long detours, and emissions from fuel consumption will be reducedas well. We assume that each CM task must be executed within the period it was revealed. Forall CM scenarios, we compute the percentage of CM tasks that could be performed within thedemanded period. The service level, defined as the average percentage of succeeded CM tasks overall scenarios, indicates how well the fleet performs under uncertainty of failures leading to CM tasks.

Of course, the performance of the fleet depends on the buffer for CM hours that was given as inputto the first subproblems. Therefore, the first and second subproblem will be solved for differentbuffer choices as well, and the service level of each resulting fleet composition will be computedagain. The size of the buffer for CM hours determines the size of the fleet composition, and conse-quently the total maintenance cost. The smaller the buffer, the lower the costs, but the higher theprobability that not all maintenance tasks can be performed within time. Based on the service levelof each fleet composition, the decision-maker should make a trade-off between cost and certainty ofcompleted tasks.

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Besides determining the fleet utilization and analyzing the performance of this fleet through thesecond and third subproblem, we will perform a sensitivity analysis as well. This will be done byvarying two other input parameters, as indicated in Figure 2: the number of annual PM hoursdemanded and the weather scenario. This will give the decision-maker insights in how a fleetcomposition is affected by changes in parameters that may turn out different than anticipated atthis moment.

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5 Computational study

The models described in Section 4 have been implemented and tested on a real FCDP problem facedby the NAM. The description of input data is described in Section 5.1. The results for differentparameter setting are presented in Section 5.2. In order to get insights in a robust solution for theFCDP, we will compare outputs for different CM buffer choices, weather scenarios and PM demand.The models are implemented in Python and solved by Gurobi. We ran all computations on a onenode Intel Xeon, 2.5 GHz of the Peregrine cluster of the University of Groningen [31].

5.1 Description of input data

The data described in this section is provided by the NAM. We consider the one year FCDP forof their offshore gas installations. The installations are located in the North Sea, between the coastof the Netherlands and the UK. Annual demand for preventive maintenance is estimated to take

. The relevant characteristicsof these gas installations are provided in Table 5. Each installation allows a maximum number ofpersonnel on board (POB), varying between Some gas installations need multiple visitsduring the planning horizon, because several maintenance tasks need to be carried out regularly.The installations owned by the NAM need to be visited times a year.

Table 5: Characteristics of offshore gas installations included in the problem.

Total Min. Max. Average

Nr. gas installationsLocated in the UKLocated in the NLInstallations with helicopter deckAnnual PM demandMax. POBVisit frequency

An overview of the transport types considered is provided in Table 6. This table summarizes, foreach transport type, the number of people on board (POB), their daily capacity, daily cost andoperational limits. Our model assumes that all transport types are available throughout the entireyear. However, weather conditions limit the availability of a transport type. This depends on theobserved weather state and the operational limits given in Table 6. In Section 4.2, we explained

Table 6: Characteristics of transport types included in the problem.

Transport POB Max. capacity Daily cost Mobilization cost Operational limit Operational limittype per day (hours) (×1000 e) (×1000 e) wind state wave state

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Table 7: Transition probability matrix for wavestates in the winter.

Wave 1 2 3state

1 0.86 0.13 0.012 0.53 0.44 0.033 0.31 0.61 0.08

Table 8: Number of days a transport type isunavailable in one year due to weather condi-tions.

Nr. of days unavailable

how to obtain relevant and reliable weather scenarios, where we followed the procedure presentedby Halvorsen-Weare & Fagerholt (2017). By using Markov chains obtained from historical dataon wave heights, we created wave state transition probability matrices for each season separately(Rijkswaterstaat, Platform K13a, 2011-2019). These transition matrices calculate the probabilityof moving from one wave state to another each day. Table 7 provides transition probabilities for thewinter. As an example, given that wave state 1 is observed today, the probability that tomorrow’swave state is 2 equals 0.13. Based on the transition matrices of all four seasons, we generated awave state scenario for one entire year. To preserve the strongly correlated wind states, we createda wind state scenario from the generated wave state scenario given their correlation. The last twocolumns in Table 6 denote the operational limit in which the transport types may operate. Forinstance, when we observe a wave state higher than 1, OSV1 and OSV2 may not operate that day.Table 8 reports how many days each type is unavailable due to bad weather in the simulated oneyear weather scenario. We observe that the vessels are affected by weather conditions more oftenthan the helicopter and barge.

The capacities per transport type, showed in Table 6, indicate the maximum number of mainte-nance hours they can perform each day. However, not all capacity can be used for conductingmaintenance tasks, because part of the capacity is lost by traveling from and to gas installations.To leave sufficient time for traveling, we incorporate a buffer that reduces the capacity of eachtransport type. By observing interim results of the second subproblem, we saw that reservingof the capacities of the vessels, and of the helicopter’s capacity, is enough to capture traveltime. The helicopter needs to travel back and forth to the harbor everyday, hence more capacityreservation is needed for this type. In addition, the vessels loose part of their capacity when theyapply dual manning. That is because part of the crew starts working later on those days, becausethe other half of the crew must be dropped of at another location first. To capture capacity lossdue to dual manning, we reduce the vessel’s capacities by another .

Besides reserving capacity for traveling, we also reserve capacity for unexpected corrective mainte-nance (CM) tasks, as explained in Section 4.2. A suitable choice for this buffer is hard to predict onbeforehand. Therefore, we will compare the results for different CM buffer choices in Section 5.2.For each CM buffer choice, we compute the total maintenance cost and corresponding service levelindicating the average fraction of succeeded CM tasks over all CM scenarios. The buffer that leadsto a fleet composition and deployment with lowest cost and a service level of 1 will be considered thebest choice. A CM buffer leading to a service level of 1 is desired, because the model must be able

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Table 9: Inter-arrival times of corrective maintenance (CM) tasks for the three gas installationtypes considered.

Gas installation type Nr. of installations Nr. of CM tasks per year Inter-arrival time

to find robust solutions. Hence, when other (worse) conditions are given as input, the model muststill find a solution with a high service level. In Section 4.5, we explained how the CM scenarios aregenerated by using a Poisson process. For each installation g, we are given the expected numberof CM tasks (Qg) during one planning horizon. The inter-arrival time λg of each installation gcan be computed as T

Qg, where T = 365. The NAM categorizes their gas installations into simple,

medium and complex. They have observed that, on average,maintenance tasks are required for . Table 9 provides an overview ofthe inter-arrival times corresponding to their gas installation type.

5.2 Results

In the following, we present and discuss results of the three models we developed for each subprob-lem. For different parameter settings, the model corresponding to the first subproblem (MILP 1)will be solved. The solutions that we obtain are evaluated right down to their performances mea-sured via the second and third subproblem. In this way, the total cost and service level of the fleetcomposition and deployment are computed and compared for the different parameter settings. Weaim at finding a suitable CM buffer choice that is given as input to MILP 1. Therefore, we start byevaluating different CM buffer settings by studying their corresponding service levels. Thereafter,we perform two sensitivity analyses by comparing the output for distinct scenarios of weather andpreventive maintenance demand.

5.2.1 Homogeneous CM buffer choices

We aim at finding a CM buffer that will result in a fleet composition that is able to perform allunpredictable CM hours next to the planned PM hours. We start by setting this buffer (the per-centage of capacity reserved for CM tasks) equal for the helicopter and vessels. The barge cannotperform corrective maintenance, hence no capacity has to be reserved for CM tasks for this modeof transport. Given the distribution of CM tasks per installation in Table 9, we expect to havearound CM tasks per year. The CM tasks faced by the NAM require on average a full day ofwork. Based on initial results from the first subproblem, we observe that together, the transporttypes are deployed around days per year. Hence, we expect that a CM buffer of =0.176 (17.6%) will result in a service level close to one.

As a first step, we investigate service levels for a broad range of buffer parameters around this num-ber, from 10% to 25% with steps of 5%. Adding the barge to subproblem 1 increases the computingtime of the MILP substantially, so we report results without a barge as well. Table 10 summarizesthe results of both cases. Reported in this table are the CPU times in seconds, the optimality

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Table 10: Results of MILP 1 and service levels obtained from solving subproblem 2 and 3.

CM buffer Inc./Excl. CPU time Optimality gap Cost objective (UB) Nr. of days scheduled Servicebarge MILP 1 (s) MILP 1 (%) MILP 1 (million e) (Heli, OSV1, OSV2, Barge) level

gap, the total cost (upper bound for the objective function) for the fleet composition resulting fromMILP 1 and the corresponding service level. We observe that, when the barge is not included inthe set of transport types, MILP 1 is able to find solutions with an optimality gap of less than 1%within a few minutes or even seconds. However, running time drastically increases when a barge isadded to the set of possible transport types. This is due to the fact that constraints (10) until (14)in MILP 1, constructed especially for the barge, add much complexity to the model. In the lastcolumn, we observe that a service level of 1 could only be reached for CM buffers of 20% and 25%.However, after simulating 1000 CM scenarios, at least 27 days of capacity remains unused for thesebuffer choices. We aim at finding a CM buffer choice that has a service level of 1, without havingscheduled too much transport types. Based on the last column in Table 10, we suspect to have amore reasonable service level when the CM buffer lies between 16% and 19%.

The computational results for buffer choices between 15% and 20% are provided in Table 11.Reported in this table are the total cost (upper bound for the objective function) for the fleetcomposition resulting from MILP 1, the corresponding service level, the ratio of CM deploymentand the number of days a scheduled transport type was not deployed. Again, we ran MILP 1 withand without the barge. The solutions with lowest cost are reported in this table. For the bufferchoices 15% until 19%, a better solution with barge in the optimal fleet composition could notbe found within a CPU time of 40 hours. Only for a CM buffer of 20% we could find a lowestcost objective within 40 hours containing a barge in the solution, which was slightly better thanthe solution without barge. We conclude from this that deploying the barge does not significantlydecrease total maintenance costs.

Table 11: Detailed results of equal buffer choices for the helicopter and Offshore Supply Vessels.

CM Cost objective (UB) Nr. of days scheduled Service Ratio CM Nr. of daysbuffer MILP 1 (million e) (Heli, OSV1, OSV2, Barge) level deployment not deployed

As expected, we observe in Table 11 that the total cost increases when the CM buffer increases.The reason for this is that more capacity is reserved for CM tasks, so less capacity is left for theplanned PM tasks. In turn, more transport types have to be deployed to maintain all PM tasks inMILP 1. Interestingly, the service level slightly decreases when the CM buffer increases from 16%to 17%. To understand this, take a closer look at the columns ‘Nr. of days scheduled’ and ‘RatioCM deployment’ in Table 11. The CVRP we solved for subproblem 2 ensures that most PM tasksare performed by the OSVs, because they can perform more PM hours per day than the helicopter,with relatively lower cost. As a result, the vessels can devote less time to CM tasks than a heli-copter. Therefore, we observe in the column ‘Ratio CM deployment’ that a helicopter is deployedfor CM tasks more often than the vessels. However, the number of days a helicopter is scheduleddecreases with 13 days when the CM buffer increases from 16% to 17%. Instead, OSV2 is scheduledfor one more period. As a result, less CM tasks can be assigned to a helicopter. Consequently,more CM tasks need to be performed by an OSV. We assumed that each CM task requires one dayof work, independent of the transport type utilized. More capacity is lost when OSV1 or OSV2performs a CM task, because they can utilize respectively 3 and 6 times more PM hours per daythan a helicopter. In conclusion, having fewer helicopters scheduled results in assigning more CMtasks to vessels, leading to a decrease in service level. The same effect appears when the bufferincreases from 18% to 19%: both buffer choices lead to a service level of 1, but the number of notdeployed days decreases.

The fleet composition and deployment with lowest cost and a service level of 1 can be found if wehomogeneously choose a CM buffer of 18%. As mentioned above, the helicopter is deployed for CMtasks more often than the vessels. Therefore, a better choice might be to reserve more capacityfor CM tasks for the helicopter than for the vessels. This will be investigated in the next subsection.

Note that our model for the CVRP was not able to find the optimal routes in all periods. We ranthe model for 10 minutes per period, leading to a CPU time of 15600 seconds for all 26 periodstogether. The optimality gaps varied from 0% to 48.93% in the worst case. We observed thatthe higher gaps correspond to periods in which more installations (e.g. 16-24 installations) requirePM tasks. So our model for the CVRP performs worse when cost-minimizing routes need to becomputed for larger problem instances. As a result, the service levels provided in Tables 10 and 11might be an underestimation.

5.2.2 Heterogeneous CM buffer choices

In the previous subsection, we started of by setting the CM buffer equal for the helicopter andvessels. It turned out that a CM buffer of 18% was the most suitable choice. In this section, wewill study the total cost and service levels for heterogeneous CM buffer settings. We do this byincreasing the CM buffer for the helicopter step by step, while decreasing the CM buffer of a vessel,starting from 18%. We disregard the barge in these computations, because for this buffer choice, abetter solution with barge could not be obtained within a CPU time of 40 hours. Figure 3 shows thecost objective and corresponding service levels for the chosen CM buffer combinations. The differentCM buffer combinations are given on the horizontal axis, and are selected based on the transporttype capacities given in Table 6.

To keep an overall CM buffer of approximately the same size as

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Figure 3: Cost objectives and corresponding service levels of heterogeneous CM buffer choices. Forcomparison, the results for the homogeneous buffer of 18% are added.

the homogeneous combination (18%, 18%, 18%), we add respectively to the helicopter’sbuffer when we subtract of the OSV1 and OSV2 buffer. On the vertical axis, we plotted the costobjective of MILP 1 and the service level computed for the corresponding optimal fleet composition.

From Figure 3, we observe that the total maintenance cost decreases when the helicopter reservesmore capacity for CM tasks. The reason for this is as follows. Reserving more capacity for CMtasks leads to the fact that less capacity is left for PM tasks. As a result, MILP 1 schedules thehelicopter less often for higher CM buffer choices, as can be obtained in Figure 4. (Note that this

Figure 4: The number of days the helicopter is scheduled in MILP 1 for the different heterogeneousCM buffer choices.

35

graph is not monotonically decreasing, probably due to optimality gaps being nonzero.) Becausethe helicopter is more expensive than the vessels (in proportion to its capacity), scheduling lesshelicopters leads to a decrease in cost. Therefore, the blue line in Figure 3 decreases when the CMbuffer for the helicopter increases. Previously, we saw that scheduling less helicopters leads to lowerservice levels. Therefore, the green line in Figure 3, representing the service level, decreases. Theincrease in the service level for the last observation is caused by the increase in scheduled helicopterdays for the same CM buffer in the graph of Figure 4.

An important insight that Figure 3 provides is that the best CM buffer combination (resulting ina solution with the highest service level and the lowest cost) is to reserve 24% of the helicopter’scapacity, 18% of OSV1, and 17% of OSV2. This combination leads to a solution having a servicelevel of 1 and a cost reduction of . The corresponding optimal fleet compositionand deployment is depicted in Figure 5. We observe that the vessels are deployed for the mostpart of the year: 22 and 23 periods out of the 26 periods. The periods in which the vessels are notdeployed concerns periods in which more than 4 days their operational weather limit is exceeded(during fall and winter). The helicopter is deployed for 130 days during one year. When the NAMschedules the transport types according to these findings, it turns out to be most beneficial to usethe vessels mostly for PM tasks, and the helicopters primarily for CM tasks. Of course, this solutionis retrieved under the assumption that the weather of the coming year is based on averages of thepast, and that maintenance demand turns out the be as currently estimated by the NAM. To gaininsights into the effect of changing those assumptions, we will analyze the performance of solutionsunder different scenarios of weather and demand in the next paragraphs. In order to achieve areasonable high service level under other circumstances, we take as input in further computationsthe heterogeneous buffer of 24%, 18% and 17% for respectively the helicopter and both vessels.

Figure 5: Optimal fleet composition and deployment for the FCDP, assuming an average weatherscenario, the expected maintenance demand and given a CM buffer of (24%, 18%, 17%) for (Heli-copter, OSV1, OSV2).

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5.2.3 Weather impact

Weather conditions affect the transport type’s maintenance deployment at offshore gas installations.The modes of transport may not execute maintenance operations when either wind speed or waveheight is above the operational thresholds as listed in Table 6. Consequently, the optimal fleetcomposition and deployment depends on the weather scenario that is given as input. The previouspresented results were obtained under a one year weather scenario based on averages of the pastdecade. Only when future weather on the North Sea behaves like in the past 10 years, the abovepresented solution is likely to perform well in practice. However, when there is reason to believethat in the coming years weather circumstances around the gas field will change, a different solutionmust be found. Due to for example climate change, future wind speed and wave height mightbecome more stable or unstable. To observe the effect of a climate transition on the optimal fleetcomposition, we computed solutions under worse and better weather circumstances. We createdfour additional weather scenarios in which wind and wave states 2 and 3 are reached respectively2, 1.5, 0.75 and 0.5 as often as in the expected weather scenario. Table 12 summarizes for eachweather scenario the number of days the transport type become unavailable per year.

Table 12: Number of days a transport type is unavailable per weather scenario.

Transport type Number of days unavailable per yearWorst-case Bad-case Expected-case Good-case Best-case

For each weather scenario, we computed the optimal fleet composition and corresponding cost re-sulting from the first model. The results are depicted in Figure 6. Recall that scheduled vesselsneed to be paid for the full period of 14 days, independent of the number of days it performs main-

Figure 6: Optimal fleet composition and deployment and corresponding total cost for differentweather scenarios.

37

tenance during that period. As a result, when future weather conditions deteriorate, the number ofdays a vessel is not deployed in a period increases. To complete all maintenance hours, the vesselsneed to be scheduled for more periods, or an alternative (more expensive) transport type needs tobe scheduled instead. Therefore, the cost increases when weather conditions become worse, as canbe obtained from Figure 6.

Figure 6 also indicates that the effect of worse weather scenarios have a larger impact on total coststhan the effect of better weather scenarios. This can be explained considering that MILP 1 has apriori information on weather circumstances. The model anticipates by deploying the vessels onlyduring weeks with good weather conditions. Hence, under the good-case and best-case weather sce-narios, there is not much room for improvement in the scheduling of OSV1 and OSV2, comparedto the scheduling of vessels under the expected weather scenario. However, under the bad-caseand worst-case scenario, more often an alternative (more expensive) mode of transport needs to bedeployed for maintenance hours that otherwise would have been performed by the vessels. Figure6 shows that the helicopter is scheduled more often, and that even the barge is chartered for afew days. Note that the effect of worse weather conditions on total costs is presumably smallerthan Figure 6 illustrates. That is due to the barge being part of the two solutions, which resultsin larger optimality gaps in the solution of MILP 1 than in the solutions calculated for the otherthree scenarios.

What can be concluded from the above, is that a climate transition has major impact on theoptimal fleet composition, especially when that climate change leads to worse weather conditionsthan currently faced. When in future years weather circumstances do not change much, the resultsunder the expected weather scenario can be implemented. When there is reason to believe thatfuture weather will behave differently, or when the NAM wants to rely on a solution for a morepessimistic weather scenario, they could implement the fleet composition that was found under thebad-case scenario. In that case, our model suggests to schedule the helicopter more often.

5.2.4 Effect of maintenance demand

Not only weather conditions affect the optimal fleet composition and deployment, also the demandfor maintenance influences the number of days each transport type needs to be deployed. Currently,the NAM estimates to yearly have PM tasks requiring a total of hours, as shownin Table 5. In the future, demand for maintenance may change. To get an insight in how mainte-nance demand affects the optimal solution, we computed the results under six other maintenancescenarios. Figure 7 shows the optimal fleet composition and deployment and corresponding totalcost for when PM and CM demand decreases or increases with 5%, 10% and 15%. The solutionsare again computed under the expected weather scenario. The figure indicates that total costs areproportional to the required maintenance hours. It verifies that our model behaves according to ex-pectations: the higher the demand, the larger the fleet composition, and hence the higher the cost.We observe that the number of days the helicopter is deployed, increases every time the demandincreases with 5%. The same applies to OSV2, although the barge seems to take over PM tasksfrom the vessels when demand increases with 10 or 15%. Under these two scenarios, the barge onlyperforms maintenance at installations with a demand of over hours and installations with avisit frequency

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6 Conclusion

In this thesis, we introduced the Fleet Composition and Deployment Problem (FCDP) for mainte-nance operations at offshore gas installations. In the FCDP, the objective is to determine a fleetcomposition and deployment that minimizes long-term maintenance costs, such that all mainte-nance tasks can be performed within the planning horizon. This problem integrates three types ofdecisions, which cannot be optimized jointly without resulting into computational unsolvable mod-els. Therefore, we divided the problem into three solvable deterministic subproblems and developeda model for each subproblem. Our first model determines the cost-minimizing fleet composition tobe used for both preventive maintenance (PM) and corrective maintenance (CM) tasks. Deploy-ment decisions are made in the second and third subproblem thereafter. For these two subproblems,we proposed a second and third model that respectively assign PM tasks and CM tasks to the fleetobtained from the first subproblem.

To deal with the uncertainty of failures leading to CM tasks, we incorporated a buffer into ourfirst model. This was done by reserving part of the transport type’s capacity for CM tasks. Conse-quently, less capacity is left for the planned PM tasks, resulting in the fact that our model schedulesthe transport types more days than needed for the given PM demand. In this way, unexpectedCM tasks can be assigned to transport types that were not yet deployed. By using such buffer,uncertainty is taken care of indirectly and the approach remains tractable. Using this methodology,cost-minimizing tactical decisions are made by the first model. These decisions can be evaluatedafterwards by observing the number of CM tasks that could be performed under simulated CMscenarios in the third model.

Our models have been implemented and tested on a real FCDP problem faced by the NAM. Wehave seen that our models were able to find good results for different parameter settings. Underthe expected scenario of weather and maintenance demand, we found a heterogeneous CM bufferleading to a service level of 1 that reduces total cost by with respect to the best ho-mogeneous CM buffer that was established. The way in which the CM buffer is incorporated in themodel, gives the NAM the opportunity to implement a solution that gives them the service level oftheir choice, where the output of the model clearly indicates the corresponding cost. In this way,the NAM is able to make trade-off decisions between cost and certainty of completed CM tasks.With regard to future expectations about annual required maintenance hours, we have seen thatour model is able to find solutions that reduce total cost by at least decreasein maintenance demand. The results presented in this thesis, have given the NAM insights in thecost and performance of fleet compositions under different circumstances. They can extent ourresearch by modifying input parameters of their choice, because the models we developed are easilyadaptive. Based on that, and based on the results of this thesis, the NAM is able to make costreducing tactical decisions for the coming years.

In this thesis, we presented a pragmatic approach for a difficult fleet composition problem. Ourmodels were successfully implemented and tested on a real size FCDP. In conclusion, this thesisprovides a new methodology for integrally scheduling maintenance operations that can be used forreal size offshore fleet composition problems.

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7 Discussion

In this section, limitations of the three models we developed are discussed and recommendationsfor further research are given. We end this section by suggesting research topics that could be ofinterest for the NAM specifically.

7.1 Model development

Regarding the model developed for the first subproblem, we observed that running time drasticallyincreases when a barge is added to the set of possible transport types. Hence, in cases where abarge is competitive with the other transport types, and the size of the problem is comparable tothe case study we conducted, it is key to find a way to reduce running time of the model. Thiscould be done either by modeling the barge constraints more efficiently, or by developing a heuristicthat finds suboptimal solutions with lower CPU times.

A second improvement of the model for the FCP could be to incorporate a buffer for correctivemaintenance (CM) tasks by periodically adding ‘slack hours’ (estimated hours needed for CM tasks)to the annual preventive maintenance (PM) hours, instead of reducing capacity per transport type.These slack hours are then added to the planned PM hours. In this way, the model ensures havingscheduled enough transport types for both type of maintenance tasks. Instead of finding the rightcombination of buffer parameters for each transport type separately, this approach only requiresfinding one suitable parameter. One important thing to note is that, by incorporating a buffer forCM tasks, uncertainty is taken care of indirectly. Hence, decisions that follow from each model donot directly take into account uncertainty. The largest limitation of this study is that it has notbeen proven that this method performs as good as stochastic programming approaches.

To solve the CVRP for larger problem instances, one could add a constraint to the model we devel-oped for the first subproblem that limits the number of installations to be maintained in one period.A second option is to separately solve the FCDP for installations in the Netherlands and the UK.In this way, the problem is solved for two instances instead of one instance with gas installa-tions. A last option is to use column generation techniques, like the ones proposed by Kallehaugeet al. (2005) and Farham et al. (2018), that can solve large instance vehicle routing problems moreefficiently.

In the third subproblem, we inserted CM tasks into the route of an available transport type thatis closest to the gas installation requiring corrective maintenance. Another approach could be toinsert CM tasks into routes where the focus lies on minimizing cost instead of minimizing traveldistance. Further research could examine the costs of interrupting a PM task for a CM operation.

Lastly, to reduce complexity of the three models, we assumed that daily capacities of all transporttypes are constant throughout the year. However, in practice, maximum capacity is often dependenton the season. Incorporating seasonality into the capacities might lead to different results and couldbe investigated in further research.

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7.2 NAM maintenance operations

We consider the following research topics that could be of interest for the FCDP faced by the NAM.

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References

[1] Akkerman, R., & Knip, M. (2004). Reallocation of beds to reduce waiting time for cardiacsurgery. Health care management science, 7(2), 119-126.

[2] Almeida R. (2014, April 28). Seafox 4. Retrieved from https://gcaptain.com/keppel-fels-workfox-discuss-plans-new-jack-rig-design/seafox-5-2/

[3] Almeida, R. (2015, February 12). A Tour of Kroonborg, Wagenborg’s New Walk-to-Work Vessel.Retrieved from https://gcaptain.com/tour-kroonborg-wagenborgs-new-walk-work-vessel/

[4] Balakrishnan, A., Ward, J. E., & Wong, R. T. (1987). Integrated facility location and vehiclerouting models: Recent work and future prospects. American Journal of Mathematical andManagement Sciences, 7(1-2), 35-61.

[5] Baldock, N., Sevilla, F., Redfern, R., Storey, A., Kempenaar, A., & Elkinton, C. (2014). Op-timization of Installation, Operation and Maintenance at Offshore Wind Projects in the US:Review and Modeling of Existing and Emerging Approaches. Garrad Hassan America, Inc., SanDiego, CA (United States).

[6] Bretthauer, K. M., Heese, H. S., Pun, H., & Coe, E. (2011). Blocking in healthcare operations:A new heuristic and an application. Production and Operations Management, 20(3), 375-391.

[7] Calvete, H. I., Gale, C., Oliveros, M. J., & Sanchez-Valverde, B. (2007). A goal programmingapproach to vehicle routing problems with soft time windows. European Journal of OperationalResearch, 177(3), 1720-1733.

[8] Cochran, J. K., & Roche, K. T. (2009). A multi-class queuing network analysis methodologyfor improving hospital emergency department performance. Computers & Operations Research,36(5), 1497-1512.

[9] Crainic, T.G. (2000). Service Network Design in Freight Transportation. European Journal ofOperational Research. 122. 272-288.

[10] Dai, L., Stalhane, M., & Utne, I. B. (2015). Routing and scheduling of maintenance fleet foroffshore wind farms. Wind Engineering, 39(1), 15-30.

[11] Damveld, H. (2015, September 2). Basiskennis aardgas en aardbevingen in 27 argu-menten. Retrieved from http://www.co2ntramine.nl/basiskennis-aardgas-en-aardbevingen-in-27-argumenten/

[12] De Kleijn, L. S. (2012). A New Offshore Access Strategy (Bachelor’s thesis, The Hague Uni-versity of Applied Sciences). Assen: Shell/NAM.

[13] Denton, B. T., Miller, A. J., Balasubramanian, H. J., & Huschka, T. R. (2010). Optimalallocation of surgery blocks to operating rooms under uncertainty. Operations research, 58(4-part-1), 802-816.

[14] Dyer, M., & Stougie, L. (2006). Computational complexity of stochastic programming prob-lems. Mathematical Programming, 106(3), 423-432.

43

[15] Farham, M. S., Sural, H., & Iyigun, C. (2018). A column generation approach for the location-routing problem with time windows. Computers & Operations Research, 90, 249-263.

[16] Gundegjerde, C., Halvorsen, I. B., Halvorsen-Weare, E. E., Hvattum, L. M., & Nonas, L. M.(2015). A stochastic fleet size and mix model for maintenance operations at offshore wind farms.Transportation Research Part C: Emerging Technologies, 52, 74-92.

[17] Gutierrez-Alcoba, A., Hendrix, E. M., Ortega, G., Halvorsen-Weare, E. E., & Haugland, D.(2019). On offshore wind farm maintenance scheduling for decision support on vessel fleet com-position. European Journal of Operational Research, 279(1), 124-131.

[18] Halvorsen-Weare, E. E., & Fagerholt, K. (2017). Optimization in offshore supply vessel plan-ning. Optimization and Engineering, 18(1), 317-341.

[19] Halvorsen-Weare, E. E., Gundegjerde, C., Halvorsen, I. B., Hvattum, L. M., & Nonas, L. M.(2013). Vessel fleet analysis for maintenance operations at offshore wind farms. Energy Procedia,35, 167-176.

[20] Hoff, A., Andersson, H., Christiansen, M., Hasle, G., & Løkketangen, A. (2010). Industrialaspects and literature survey: Fleet composition and routing. Computers & Operations Research,37(12), 2041-2061.

[21] Hofmann, M. (2011). A review of decision support models for offshore wind farms with anemphasis on operation and maintenance strategies. Wind Engineering, 35(1), 1-15.

[22] Hulshof, P. J., Kortbeek, N., Boucherie, R. J., Hans, E. W., & Bakker, P. J. (2012). Taxonomicclassification of planning decisions in health care: a structured review of the state of the art inOR/MS. Health systems, 1(2), 129-175.

[23] International Energy Agency (2020, Sep 16). Gas - Fuels & Technologies. Retrieved fromhttps://www.iea.org/fuels-and-technologies/gas

[24] Irawan, C. A., Ouelhadj, D., Jones, D., Stalhane, M., & Sperstad, I. B. (2017). Optimisation ofmaintenance routing and scheduling for offshore wind farms. European Journal of OperationalResearch, 256(1), 76-89.

[25] Jonker, T. (2017). The development of maintenance strategies of offshore wind farm. Literatureassignment ME54010.

[26] Kallehauge, B., Larsen, J., Madsen, O. B., & Solomon, M. M. (2005). Vehicle routing problemwith time windows. In Column generation (pp. 67-98). Springer, Boston, MA.

[27] Larsen, E., Lachapelle, S., Bengio, Y., Frejinger, E., Lacoste-Julien, S., & Lodi, A. (2019).Predicting Tactical Solutions to Operational Planning Problems under Imperfect Information.

[28] Maisiuk, Y., & Gribkovskaia, I. (2014). Fleet sizing for offshore supply vessels with stochasticsailing and service times. Procedia Computer Science, 31, 939-948.

[29] NHV Group (2020, July 1). NHV awarded offshore contract from Shell U.K. and NAM. Re-trieved from https://verticalmag.com/press-releases/nhv-awarded-offshore-contract-from-shell-u-k-and-nam/

44

[30] Patout, P. J., & Schellstede, H. J. (2002). Movable self-elevating artificial work island withmodular hull. U.S. Patent No. 6,499,914. Washington, DC: U.S. Patent and Trademark Office.

[31] Peregrine HPC cluster, https://www.rug.nl/society-business/centre-for-information-technology/research/services/hpc/facilities/peregrine-hpc-cluster

[32] Premkumar, P., & Kumar, P. N. (2020). Locomotive assignment problem: integrating thestrategic, tactical and operational level aspects. Annals of Operations Research, 1-32.

[33] Rademeyer, A. L., & Lubinsky, D. J. (2017). A decision support system for strategic, tacticaland operational visit planning for on-the-road personnel. South African Journal of IndustrialEngineering, 28(1), 57-72.

[34] Redmer, A., Zak, J., Sawicki, P., & Maciejewski, M. (2012). Heuristic approach to fleet com-position problem. Procedia-Social and Behavioral Sciences, 54, 414-427.

[35] Schrotenboer, A. H., uit het Broek, M. A., Jargalsaikhan, B., & Roodbergen, K. J. (2018).Coordinating technician allocation and maintenance routing for offshore wind farms. Computers& Operations Research, 98, 185-197.

[36] Schrotenboer, A. H., Ursavas, E., & Vis, I. F. (2020). Mixed Integer Programming models forplanning maintenance at offshore wind farms under uncertainty. Transportation Research PartC: Emerging Technologies, 112, 180-202.

[37] Smith-Daniels, V. L., Schweikhart, S. B., & Smith-Daniels, D. E. (1988). Capacity managementin health care services: Review and future research directions. Decision Sciences, 19(4), 889-919.

[38] Stalhane, M., Halvorsen-Weare, E. E., Nonas, L. M., & Pantuso, G. (2019). Optimizing vesselfleet size and mix to support maintenance operations at offshore wind farms. European Journalof Operational Research, 276(2), 495-509.

[39] Stalhane, M., Hvattum, L. M., & Skaar, V. (2015). Optimization of routing and scheduling ofvessels to perform maintenance at offshore wind farms. Energy Procedia, 80, 92-99.

[40] Stalhane, M., Vefsnmo, H., Halvorsen-Weare, E. E., Hvattum, L. M., & Nonas, L. M. (2016).Vessel fleet optimization for maintenance operations at offshore wind farms under uncertainty.Energy Procedia, 94, 357-366.

[41] Testi, A., & Tanfani, E. (2009). Tactical and operational decisions for operating room planning:Efficiency and welfare implications. Health Care Management Science, 12(4), 363-373.

[42] Vis, I. F., de Koster, R. M. B., & Savelsbergh, M. W. (2005). Minimum vehicle fleet size undertime-window constraints at a container terminal. Transportation science, 39(2), 249-260.

[43] Wagenborg. Purpose built walk to work vessels. Retrieved from https://www.wagenborg.com/offshore-towage/walk-to-work-services

[44] Wu, T. H., Low, C., & Bai, J. W. (2002). Heuristic solutions to multi-depot location-routingproblems. Computers & Operations Research, 29(10), 1393-1415.

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