# Decision Support Models for Operations and Maintenance for Offshore Wind Farms: A Review
## 5 Optimization  
Optimization is the mathematical concept of finding the optimal solution to a (utility) function, given certain constraints to the solution. In the case of maintenance scheduling, the most obvious goal is to minimize the costs. Typical constraints are the availability of vessels, technicians and spare parts, environmental constraints, such as wave height and wind speeds and legal constraints, such as working hours, vessel speeds or limitations to routing. In mathematical optimization, one can distinguish between single- objective optimization and multi-objective optimization. The latter is the case when there is not one single goal for the optimization, but multiple. In the case of the offshore \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Publication** & **Monte Carlo** & **Discrete Events** & **[PERSON]** & **Others** \\ \hline [PERSON] and [PERSON] [46] & ✓ & ✗ & ✗ & ✗ \\ [PERSON] et al. [80] & ✓ & ✓ & ✗ & ✗ \\ [PERSON] et al. [80] & ✓ & ✓ & ✗ & ✗ \\ [PERSON] et al. [81] & ✓ & ✓ & ✗ & ✗ \\ [PERSON] et al. [82] & ✓ & ✓ & ✗ & ✗ \\ [PERSON] et al. [30] & ✓ & ✓ & ✓ & ✗ \\ [PERSON] et al. [12] & ✓ & ✓ & ✓ & ✗ \\ [PERSON] et al. [14] & ✓ & ✓\({}^{\,1}\) & ✗ & Weibull distribution, autoregressive model 2 \\ [PERSON] and [PERSON] [16] & ✓ & ✓ & ✓ & binomial process \\ [PERSON] and [PERSON] [83] & ✗ & ✗ & ✗ & Petri nets 3 \\ [PERSON] et al. [17] & ✓ & ✓ & ✗ & Poisson process, agent-based method \\ [PERSON] et al. [20] & ✓ & ✗ & ✗ & multivariate autoregressive model 2 \\ [PERSON] and [PERSON] [21] & ✓ & ✓ & ✗ & Poisson process, agent-based method \\ [PERSON] et al. [33] & ✓ & ✗ & ✗ & Business Process Model Notation 4 \\ [PERSON] et al. [22] & ✓ & ✗ & ✗ & multi-agent system 5 \\ [PERSON] and [PERSON] [24] & ✗ & ✗ & ✗ & stochastic model \\ [PERSON] et al. [25] & ✗ & ✗ & ✗ & mixed-integer solver, rolling-horizon heuristic \\ [PERSON] et al. [26] & ✓ & ✗ & ✗ & ✗ \\ [PERSON] and [PERSON] [41] & ✗ & ✗ & ✗ & genetic algorithm \\ \hline \hline \end{tabular}  
* [PERSON] et al. [14] use a Bayesian belief network, which is an event model, displayed as an acyclical graph, combined with tables of condition probability. [PERSON] [84] introduces the concept. 2 An autoregressive model is a special case of the more general ARIMA model for time series, explained in e.g., [PERSON] and [PERSON] [85]. 3 A Petri net is a discrete event dynamic system in graphical notation, see [PERSON] [86] for an introductory review. For limited states it can be converted to a direct graph [87]. 4 Business process model notation is a type of graphical modeling notation. [PERSON] et al. [88] provide a literature review and classification of business process modeling, while [PERSON] and [PERSON] [89] review the notation. 5 In multi-agent systems multiple agents act and react upon their environment and other agents. [PERSON] and [PERSON] [90] present a review of multi-agent systems in manufacturing and supply chain management.  
\end{table} 
Table 4: This table summarizes the kinds of models presented in the literature. wind farm maintenance, this can mean to not only minimize costs, but at the same time maximize availability of the wind farm. This seems to be equivalent at the first glance, since downtime leads to production losses. However, maximizing availability will lead to a higher number of corrective maintenance actions that cannot be grouped together. This in turn influences the cost of maintenance. Legal restrictions and public opinion can be reasons for the wind farm operator to maximize availability, while still trying to minimize costs. In the following, we give an overview over different mathematical optimization algorithms and methods. The overview includes existing optimization techniques, with a focus on those that are implemented in the state-of-the-art maintenance models. For a detailed introduction to optimization, we want to refer to literature in the field, such as [PERSON] [91]. Many introductory books on optimization focus on the type of problem (model) for which an optimal solution is found.