# Developments, Trends, and Challenges in Optimization of Ship Energy Systems
## 4 Historical Development of Optimization of Ship Energy Systems
### Optimization Algorithms  
The continuous increase of the interest in applying optimization and of the complexity of optimization problems leads to a continuous improvement of existing and development of new optimization methods and algorithms. The algorithms used in the articles of Table 1 are presented in Table 5.  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Objective Function** & **Articles** \\ \hline Weighted summation of (i) energy efficiency, (ii) exergetic efficiency, & 52 \\ (iii) unit cost of fresh water & 53 \\ \hline Weighted summation of (i) energy efficiency, (ii) W\({}_{\text{net}}\)/UA & 53 \\ \hline (i) Energy and exergy efficiencies (max), (ii) system investment/payback period (min) & 55 \\ \hline (i) Average voyage cost (min), (ii) GWP emissions (min) & 57 \\ \hline Biobjective optimization: two of the following objective functions are selected for each one & \\ of six optimization problems: (i) lifetime fuel consumption, (ii) overall propulsive efficiency, & 37 \\ (iii) total ship resistance, (iv) GHG emissions across the entire operating range, (v) total cost & \\ \hline \hline \end{tabular}
\end{table}
Table 4: _Cont_.  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Algorithm** & **Articles** \\ \hline
**Mixed-integer linear programming algorithms** & \\ \hline Branch and Bound & 10, 11 \\ \hline Mixed Integer Linear Programming & 35 \\ \hline
**Constrained nonlinear programming algorithms** & \\ \hline Generalized Reduced Gradient in GRG2 Software & 7, 17 \\ \hline Sequential Quadratic Programming (SQP) & 3, 9, 11, 13, 18 \\ \hline Interval arithmetic implemented in INTLAB-Version 5.5 & 15 \\ \hline MINLP problem implemented in GAMS 23.6 software & 36 \\ \hline Combined use of NLopt and COBYLA & 38, 39 \\ \hline GUROBI 8.1.0 & 49 \\ \hline fmincon & 56 \\ \hline
**Dynamic optimization algorithms** & \\ \hline Sequential direct method combined with SQP & 46, 48 \\ \hline Deterministic Dynamic Programming (DDP) & 57 \\ \hline
**Stochastic or evolutionary algorithms** & \\ \hline Particle Swarm Optimization (PSO) & 1, 2, 22, 31, 32, 41, 42 \\ \hline Adaptive Multicontext Cooperatively Coevolving Particle Swarm & 28 \\ Optimization algorithm (AM-CCPSO) & \\ \hline Genetic Algorithm (GA) & 4, 5, 6, 7, 17, 18, 19, 23, 25, 27, 45, 46, 52 \\ \hline Struggle Genetic Algorithm (StrGA) & 1, 2 \\ \hline Nondominated Sorting Genetic Algorithm (NSGA-II) & 8, 30, 37, 40, 47, 50, 55, 57 \\ \hline Improved NSGA-II & 24 \\ \hline Differential evolution optimization algorithm & 44 \\ \hline Mixed Integer Distributed Ant Colony Optimization solver (MIDACO) & 33 \\ \hline Whale Optimization Algorithm & 51 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Algorithms and/or software used in the articles of Table 1.  
In Ref. [2], a list of algorithms is included that have been applied for optimization of energy systems on land. Many of those algorithms appear also in Table 5, while there are algorithms in Table 5 that do not appear in Ref. [2]. Cross-fertilization between the two similar fields (land-ship energy systems) can be very fruitful.