## 1 Introduction  
Through new technologies development, today's industry is currently experiencing the 4 th revolution, referred to as Industry 4.0. This reality has a significant impact on the industrial management processes [1, 2, 3]. Hence, the customers practices are changed, they can, at any moment, create or modify their commands through new technological tools, such as the laptop, the tablet computer, or the smart phone [4, 5]. Indeed, these situations have an impact on the production scheduling [6, 7]. In addition, production problems can also disrupt the already established sequence, for instance, the mechanical machine breakdowns, or the lack of raw materials, etc. These factors force many production firms to react for updating the existing schedule. Hence, the rescheduling is necessary to revise schedules in a cost-effective manner [8]. Rescheduling is the mechanism of updating an established production schedule as a response to disruptions [8]. Many types of disruptions have been studied, such as the arrival of new jobs [9, 10], machine breakdowns [11, 12, 13], operational unit failures [14], job rejection [15], quality problems and shortage of raw materials [16, 17]. Over the last few years, many researches dealt with this area and dedicated to it more attention. Indeed, rescheduling studies have considered many types of machine environments [18, 19, 20, 21]. The classical efficiency criteria are often used for measuring the performance of the schedule, such as the Makespan [22, 23, 24, 25], Makespan combined with tardiness [26], total completion times [27, 28], total weighted completion times [29, 30], total weighted tardiness [31], total tardiness [32], total flow time [33] and maximum lateness [34]. Different from existing works, this one investigates new performance measures. In terms of efficiency,the Total Weighted Waiting Times (_TWWT_) is considered as a criterion. The waiting time represents the duration that the job has waited in the system before being executed [35]. In some real-life environments, the _TWWT_ is a very significant criterion. For instance, in production systems, it can represent the waiting time of a job in front of a workstation, considering the importance of the job as a weight. In the operating rooms scheduling, it represents the delay between the patient's arrival and the starting time of his treatment, considering the patient emergency level as a weight. In [PERSON] et al., the authors addressed a problem of rescheduling the jobs considering as an objective the maximum waiting time. The problem is inspired from quartz manufacturing industry. In which the waiting time is considered as the waiting for certain materials in oven before the reheating step, where both the rework flow and the regular production flows meet [36]. Thus, the authors studied and analyzed the complexity of a rescheduling problem on a single machine, to decide where to insert the rework jobs in the initial sequence of regular jobs, to minimize the maximum waiting time. [PERSON] and [PERSON] considered a problem of the total waiting time minimization on a single machine rescheduling environment. The authors formulated two mixed integer programming models [37]. The studied problem is also an illustration of quartz glass factory, where the waiting time represents the waiting of materials before the welding step. The minimization of the waits saves the energy consumption. In our study, the _TWWT_ is considered as a criterion. This criterion has been discussed for the first time by [PERSON], in 1976 [38]. Even if the efficiency measures the performance of a scheduling system, it is not the only measure for dynamic rescheduling problems. Stability measure, limiting the deviation from the initial schedule is also studied in the literature [39; 40; 41]. During rescheduling event, this measure evaluates the impact of the movement of the jobs [42].