# Bi-Objective Dynamic Multiprocessor Open Shop Scheduling: An Exact Algorithm
## 5 Conclusions and Future Research  
This paper presented an exact solution algorithm for generating optimal Pareto front solutions for the dynamic multiprocessor open shop scheduling problem considering simultaneously the two objectives of minimizing the makespan and minimizing the mean weighted flow time. Both objectives are important in practical real-life applications. The developed exact algorithm is based on the \(\varepsilon\)-constraint method. It utilizes CPLEX, a state-of-the-art MILP solver, for solving subproblems generated based on a developed MILP model. Computational experiments are conducted on a set of randomly generated 30 instances to assess the computational time requirements of the developed algorithm and to investigate the characteristics of the optimal Pareto front of the studied problem. Computational time results demonstrated that the developed exact algorithm can be practically useful for instances having up to 10 jobs, five workstations, and one or two machines in a workstation. In 80% of the cases, the computational times were found to be less than 10,000 s. In some real-life applications, such a computational time limit can be acceptable if the time span of running the schedule itself is over a few days. Alternatively, instead of generating the optimal Pareto front solutions, the CPLEX solver can be terminated at a pre-specified time limit. This will result in approximate solutions that are not proven to be optimal, yet they represent a good approximation of the optimal Pareto front. This adaptation allows the utilization of the developed exact algorithm as an approximate approach. Regarding the characteristics of the optimal Pareto front of the studied problem, computational results revealed that the size of the optimal Pareto front is small as it is composed of a small number of six to seven solutions on average. No evidence is found for the correlation between the size of the problem instance represented by the number of operations and the number of optimal Pareto front solutions. This result indicates that future algorithmic approaches need to pay more attention to intensification strategies as opposed to obtaining dense non-dominated solutions. This may contradict the traditional intention of multi-objective metaheuristic approaches that look for having a large \"volume\" of non-dominated solutions. Generally, intensification can be achieved by well-designed neighborhood search functions that are capable of improving solutions in both directions of the considered objective functions. In addition, computational results show that the optimal Pareto front is more stretched in the direction of the makespan objective by having larger distances between every pair of consecutive solutions along the Pareto front compared to the mean weighted flow time objective. This characteristic may imply that more attention needs to paid to algorithmic techniques that seek to improve the MWFT objective to be able to deal with the small distances in that direction. \begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Indicator** & **Mean** & **Std. Dev.** & **Minimum** & **First Quartile** & **Median** & **Third Quartile** & **Maximum** \\ \hline \(\sigma_{\text{C}_{max}}\) & 1.913 & 2.215 & 0.214 & 0.694 & 1.350 & 1.971 & 10.510 \\ \(\sigma_{MWFT}\) & 0.5646 & 0.3687 & 0.1138 & 0.2742 & 0.4413 & 0.7204 & 1.3145 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Descriptive statistics for the spacing indicators based on 27 instances. Future metaheuristic developments for the studied problem can benefit from recent population management structures such as the use of a reference set and the adaptive hybrid mutation operation which can be effectively used to manage neighborhood search functions in evolutionary based metaheuristics as demonstrated by [PERSON] et al. (2017). This research received no external funding. The author declares no conflict of interest.