# Multi-Objective Ship Route Optimisation Using Estimation of Distribution Algorithm
## 6 Conclusions  
We have shown--on the example of optimisation of a ship's route in tidal waters--an innovative adaptation of the estimation of distribution algorithm (EDA), intended for (simulation based) multi-objective optimal path planning in non-stationary environments. The key elements of the proposed approach (apart from the specific design space) have been the adaptive Markov chain-based path generator (that directs the global search process) and the dynamic programming-based local search algorithm (that boosts--thanks to its exploitation capabilities--the global search process; see Table 2). The presented experimental results indicate not only the high effectiveness of the proposed algorithm in finding very good quality (see Figures 10 and 11) and diversified approximations (see Figures 8 and 9) of optimal--in the Pareto sense--solutions but also a good exploration--exploitation balance of the search process. The proposed algorithm can also be applied to other domains. As mentioned above, it can be seen as a framework for multi-objective optimal path planning (also simulation based) in non-stationary environments. Future research work could concentrate on the following: \begin{table}
\begin{tabular}{l|c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multicolumn{4}{c}{**EDA Only (\(E\))**} & \multicolumn{4}{c}{**EDA + Local Search (\(EL\))**} & \multicolumn{4}{c}{(\(.\))\(.\)\(=|EL-E|/E[\%]\)} \\ \cline{2-13}  & \(|PS|\) & \(HV\) & \(t_{min}\) & \(fc_{min}\) & \(d_{0}\) & \(|PS|\) & \(HV\) & \(t_{min}\) & \(fc_{min}\) & \(d_{0}\) & \(|PS|_{r}\) & \(HV_{r}\) & \(t_{min}\), & \(fc_{min_{r}}\) & \(d_{0r}\) \\ \hline \([Prob1]\) & 76 & 0.326 & 12.11 & 27.71 & 31.86 & 512 & 1 & 10.31 & 20.68 & 27.84 & 574 & 207 & 15 & 25 & 13 \\ \([Prob2]\) & 89 & 0.345 & 12.47 & 26.43 & 31.28 & 907 & 1 & 10.43 & 19.77 & 27.10 & 919 & 190 & 16 & 25 & 13 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the results obtained by the proposed algorithm, both without and with the local search algorithm. The last part shows the percentage improvement in results when using local search. The individual symbols have the following meanings: \(|PS|\) (size of the Pareto front), \(HV\) (HV metric value), \(t_{min}\) (minimum passage time), \(fc_{min}\) (minimum fuel consumption), and \(d_{0}\) (\(l_{2}\)-distance of the nearest point of the Pareto front from the criterion space origin). Figure 17: Same as Figure 16 but for [_Prob2_]. * Verification of the proposed algorithm with a more accurate ship simulator (based on ship dynamics);
* Verification of other (selected) existing meta-heuristics as the global search process engines;
* The search algorithm hyper-parameters tuning, combined with sensitivity analysis;
* The parallelisation of the algorithm;
* The use of other types of surrogate models. Conceptualisation, R.D. ([PERSON]); Data curation, R.D. ([PERSON]); Formal analysis, R.D. ([PERSON]); Funding acquisition, R.D. ([PERSON]); Investigation, R.D. ([PERSON]) and R.D. ([PERSON]); Methodology, R.D. ([PERSON]); Project administration, R.D. ([PERSON]); Resources, R.D. ([PERSON]); Software, R.D. ([PERSON]); Supervision, R.D. ([PERSON]) and R.D. ([PERSON]); Validation, R.D. ([PERSON]); Visualisation, R.D. ([PERSON]) and R.D. ([PERSON]); Writing--original draft, R.D. ([PERSON]) and R.D. ([PERSON]); Writing--review and editing, R.D. ([PERSON]) and R.D. ([PERSON]). All authors have read and agreed to the published version of the manuscript. This research was supported by the Polish Ministry of Science and Higher Education funds assigned to AGH University of Krakow. Not applicable. The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author. The authors declare no conflicts of interest.