## 1 Introduction
In real life applications, the changes of the sequence may generate supplementary costs, such as raw materials reordering costs and reallocation costs [43; 44; 45; 46; 47; 48; 49; 50]. In this work, a new stability criterion is studied, it consists in measuring the difference between the sum of completion times of the jobs in the initial schedule and in the new one, assigning to each job a weight, referred to as the total weighted completion time deviation (_TWCTD_). The weights association makes difficult to disrupt the jobs with high weights. This is the case of an urgent patient in a hospital, for example. [PERSON] et al. [8] and [PERSON] [51] distinguished two basic rescheduling strategies. The first strategy is completely reactive, it is often referred to as dynamic scheduling strategy. In this strategy, the schedule is dynamically constructed by dispatching jobs when it is necessary. It uses locally the available information to decide which job to execute. Therefore, dispatching rules or other heuristics are used to prioritize jobs waiting to be processed on the machine [52; 53]. The second is the predictive--reactive rescheduling strategy, it consists in firstly generate an initial schedule and to update secondly this schedule, at each time a disruption occurs [54; 55; 56; 57]. In our approach, the predictive-reactive rescheduling strategy is also adopted through firstly solving an initial problem with the objective of minimizing the _TWWT_. After the occurrence of a disruption, the objective becomes to minimize a new criterion combining simultaneously the schedule efficiency and the schedule stability, with the _TWCTD_ as a stability measure. As in the majority of the rescheduling problems considering the efficiency and the stability [58; 59], a relationship between both objective is established, by associating a coefficient for each measure (i.e., \(\alpha\) efficiency + (1 \(-\)\(\alpha\)) stability), referred to as the efficiency-stability coefficient. The influence of this coefficient on the system is later discussed. Considering simultaneously the schedule efficiency and schedule stability in the second objective function, plays a more significant role for reducing the actual costs in real applications. In the literature, some classical single machine scheduling models, uses rescheduling assumption to determine new schedules in response to disruptions. These previous models are only based on the classical measures Thus, several costs are disregard. Along with that, based on the numerical results observation, a new concept consisting in increasing the weight in function of time is introduced. This concept allows to avoid ignoring the jobs with low weightsthanks to increasing their weights in function of time. The weight increasing effect is also discussed in the numerical results section. This work investigates a single machine rescheduling problem, the disruptions are caused by new jobs arrival. The choice of the single machine environment is motivated by the fact that this environment represents several real-life cases, for example a single operating room in a hospital or a single workstation in a factory. Moreover, according to [PERSON], the single machine provides a basis for heuristics that are applicable to more complicated machine environments [60]. In practice, scheduling problems in more complicated machine environments are often decomposed into sub-problems that deal with single machines. Thus, a mixed integer linear programming model is implemented to describe this problem and an iterative methodology is proposed for dealing with the online part. This work is a contribution to operational research area. The main research objectives are:  
* Considering simultaneously the _TWWT_ as the schedule efficiency and the _TWCTD_ as the schedule stability. These two criteria are mixed in one performance measure. * Implementing for this specific problem, a mathematical model based on the MILP and an iterative predictive-reactive methodology for solving the online problem.