## 7 Conclusions and Perspectives  
In this paper, a new optimization criterion is proposed to evaluate the performance of the schedules on a single machine rescheduling problem. This new criterion combines simultaneously the total weighted waiting time as a schedule efficiency and the total weighted completion time deviation as a schedule stability. This association of criteria has never been studied in the literature before and it can be a very helpful and significant criterion in industrial and hospital environments. A MILP model is proposed and an iterative predictive reactive strategy is established for dealing with the online part. In dynamic environments, the rescheduling is often used to organize the production activities. In such situations, the customer priority has a major importance, since it defines which order to prioritize. Therefore, it was consistent to consider this factor on the performance measure. Thus, the weights are considered in the efficiency and stability criterion. However, by testing our approach, we have observed that the low weighted operations may be constantly postponed by the occurrence of new high weighted operations, the solution proposed to overcome this drawback consists in varying the weight of operations in function of the time spent in the system. In this case, the challenge will be how to regulate the weight variation so as not to impact the schedule efficiency. The established study demonstrates a large percentage deviation of the objective function from the case of the static weights, but the proposed conception helps the low weighted operations to be executed. The model resolution allows us to conclude that:  
* The increase of the _TWWT_ at each rescheduling step is due to the low-weight jobs. These jobs move forwards after each disruption since the new arriving jobs are included in the middle of the schedule.
* Increasing the weight in function of the time helps the low weight earliest jobs to be executed. However, the values of the weight index must be increased for the high values of \(\alpha\).
* Analyzing the maximum duration of iteration, the computing time depends on both the set of initial jobs and the appearance frequency of the jobs.  
This work can be of great interest for the decision makers that manage rescheduling systems in hospital or industrial environments. It can help them to provide, at each rescheduling step, an optimal schedule for a single machine problem. The main limitation of this work is the number of the jobs. Firstly, we have considered only one job arrives per period. Secondly, the MILP is limited to averagely solve 35 jobs in a horizon of 48 \(ut\) (equivalent to 8 h). At those time, even before COVID-19 disaster, the pressure on healthcare services is continuously increasing. In fact, the hospital decision makers must quickly establish a planning in response to the emergency patients' arrival, even if several ones arrive at the same time. Thus, the execution time is also an interesting debate. Heuristics _wSPT_ and FIFO have proved their speed to cope with this kind of problem, but their efficiency is weak compared the MILP resolution. Therefore, one of the main short-term prospects is to design efficient heuristics and metaheuristics to browse even more jobs in a reasonable time, as well as a comparison of the proposed approximate methods with other existing heuristics. We will also consider multi machines problems at middle-term research objective.  
Conceptualization and methodology, A.T., C.S. and N.S. MILP development, computing tests and manuscript writing, A.T. Ideas validation, work supervision, reviewing and editing the manuscript, C.S. and N.S. All authors have read and agreed to the published version of the manuscript.  
This work is financially supported by Urban Community of Sarreguemines-France and Grand Est Region-France.  
Not applicable.