# A Novel Hybrid Simulated Annealing for No-Wait Open-Shop Surgical Case Scheduling Problems
## 9 Conclusions  
This paper dealt with a novel NWOSP-SCSP case of open-shop problems under makespan minimization with the transportation times for OR as a single objective model based on MILP. Since the optimization of OR problems is an NP-hard optimization problem, we assessed mathematical and metaheuristic methods to address OR optimization problems. Problem modeling was described in the context of a multi-transportation system. Two computational experiments were considered to evaluate the performances of MILP, heuristic, SA metaheuristic, and a hybrid algorithm. In the first experiment, we generated small-sized instances by which we compared the mathematical models and evaluated the general performance of the proposed metaheuristics. In the second experiment, the potential of metaheuristics to solve some benchmarks in the literature of pure open shops was further assessed. A heuristic algorithm was created to solve problems of different sizes, and its performance was compared with the SA algorithm. The results of these two algorithms led to the presentation of a novel hybrid SA algorithm. With a hypothesis test in different quarters, its performance was evaluated. The hypothesis test is used in four parts of the algorithm's performance to assess the SA and hybrid SA algorithms. Thus, the first hypothesis test was used in all quarters of the function of the two algorithms, and in this test, the H\({}_{0}\) assumption was rejected. In other words, the two algorithms have a significant difference. After reaching the results with a numerical analysis in Nova Scotia health authority hospitals and health centers, the results of the evaluations show the remarkable superiority of the hybrid SA algorithm. Finally, all the results proved that the models and metaheuristics effectively tackled the NWOSP-SCSP approach, which will be a new perspective in future literature reviews.  
Future StudyThis research assumes that the number of surgical operations equals the number of ORs. In other words, each surgical operation is processed exactly in each OR. A situation where the number of operations differs from the number of ORs is suggested for consideration in future studies. For example, one or more surgical operations will not be processed in one or more ORs. Solving the problem with other metaheuristic algorithms and comparing them with the SA algorithm results is also suggested. In the present study, a heuristic algorithm was presented for solving \(O|TT|D\), \(MT|C_{max}\) problems and was compared with the SA algorithm, and finally, a hybrid SA algorithm was suggested to obtain the best results; some studies to find an algorithm for solving \(O|TT|D\), \(ST|C_{max}\) problems are suggested if there is single transportation (\(ST\)) between each OR. As interesting future research, one could study a multi-objective no-wait open shop in surgical case scheduling problems. Another exciting research direction is considering branch-and-bound or other exact methods. A comprehensive review of bi-objective and multi-objective OR scheduling is recommended. Finally, further research on developing novel and hybrid strategies for delivering an actual problem is suggested.  
Methodology, A.R., S.M.H., M.Z. and M.M.; conceptualization, A.R., M.Z. and M.M.; software, S.M.H.; validation, A.R.; formal analysis, S.M.H. and M.M.; investigation, M.M.; resources, A.R. and M.M.; data curation, A.R., S.M.H., M.Z. and M.M.; writing--original draft preparation, M.M.; writing--review and editing, M.M.; visualization, A.R., S.M.H. and M.Z.; supervision, M.Z.; project administration, A.R., S.M.H., M.Z. and M.M. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The data used in the study are available from the authors and can be taken upon acceptable requests.  
The authors would like to thank the reviewers and the editor-in-chief for their helpful comments and recommendations.  
The authors declare no conflict.