## 5 Conclusions  
In this study, a new algorithm is proposed for the MMO of PFSSP, i.e., provide multiple solutions with the same objective value. The proposed algorithm utilizes GA for the optimization procedure with the k-means clustering algorithm being used to cluster the individuals of each generation. Unlike standard GA, where all individuals belong to the same cluster, the individuals in the proposed approach were split into multiple clusters and crossover operator is restricted to individuals belonging to the same cluster. Doing so enabled the possibility of the proposed algorithm to find multiple global optima in each cluster. Sharing fitness and CM utilized by [PERSON] et al. for the MMO of JSSP were also modified and used for the MMO of PFSSP. The performance of the three algorithms was tested on various benchmark PFSSP and the results indicated that the proposed algorithm had a better performance than CM and sharing fitness, in terms of its ability to find the best-known optimal solution, consistently provide a better optimal solution, and converge to multiple optimal solutions.  
As the sharing fitness method had a better performance on a few problems in terms of finding more optimal solutions, an attempt was made to combine the sharing fitness method with the proposed algorithm. This was done by using the feature matrix of the proposed algorithm in the sharing fitness method. When applied for MMO of benchmark PFSSP, the new hybrid algorithm was unable to outperform or equal the proposed algorithm.  
Though the proposed algorithm had the best performance of the algorithms tested, further research needs to be performed in the following areas:  
1. Introduction of more rigorous similarity measure metrics for a more efficient clustering.
2. Comparison of the performance of the proposed approach to algorithms specifically developed for optimization of PFSSP.
3. Embedding the proposed approach in algorithms specifically developed for optimization of PFSSP and use it to achieve the MMO of PFSSP.
4. Applying the proposed approach to real-life operational environment, testing the robustness of the proposed algorithms and its parameters, and measuring its performance under different criteria.  
P.Z. and M.R., analysis of the data, development of code, and writing of the manuscript. S.Y.L., guideline for the proposed approach and comments for the paper. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Datasets used in this paper are available at [[http://mariusz.makuchowski.staffi.iar.pwr.wroc.pl/download/courses/sterowanie.procesami.dyskretmymi/](http://mariusz.makuchowski.staffi.iar.pwr.wroc.pl/download/courses/sterowanie.procesami.dyskretmymi/)]([http://mariusz.makuchowski.staffi.iar.pwr.wroc.pl/download/courses/sterowanie.procesami.dyskretmymi/](http://mariusz.makuchowski.staffi.iar.pwr.wroc.pl/download/courses/sterowanie.procesami.dyskretmymi/)) (accessed on 22 February 2021).  
**Conflicts of Interest:** The authors declare no conflict of interest.