## 4 Problem Description and Strategy  
### Problem Description  
In this paper, we study a single machine rescheduling problem with disruptions caused by the arrival of new jobs. The considered problem is solved in two phases. The phase before the disruptions concerns the _TWWT_ minimization in a static single machine scheduling problem. The second phase aims to minimize, after each disruption, simultaneously, the _TWWT_ of jobs and the _TWCTD_ of uninitiated jobs. In the considered problem, for each job \(j\) is associated a weight \(w_{j}\), a release date \(r_{j}\) and a processing time \(p_{j}\). When the processing of a job \(j\) starts, it is processed without preemption to completion time \(C_{j}\). In this phase of the problem (predictive phase), all the job information (\(p_{j}\), \(r_{j}\), \(w_{j}\)) is known in advance. Thus, the waiting time \(W_{j}\) of job \(j\) is defined as \(W_{j}=C_{j}-p_{j}-r_{j}\) (see Figure 1). _Efficiency criterion_: The waiting time represents the period that the job waits, from its release date \(r_{j}\), until the execution at the starting time \(S_{j}\), i.e., \(W_{j}=S_{j}-r_{j}\). Thus, \(W_{j}=C_{j}-p_{j}-r_{j}\), knowing that \(S_{j}=C_{j}-p_{j}\) (see Figure 1). In this system, it is supposed that \(N=\{1,\,2,\,\ldots,\,n\}\) is a set of jobs, which are already available for scheduling at the time \(t=0\). In the predictive phase, the objective is to determine an initial schedule with the objective of minimizing the _Total Weighted Waiting Times_, i.e., \(\sum_{j=1}^{n}w_{j}W_{j}\). To solve this initial problem, a mathematical model is implemented. This problem can be represented by \(1|r_{j}|\sum w_{j}W_{j}\) in the standard notation. During the initial schedule execution, a new job can arrive. Accordingly, this new job will be combined with uninitiated jobs to create a new set of jobs \(N^{\prime}=\{1,\,2,\,\ldots,\,n^{\prime}\}\) and the existing schedule is updated. The new objective is to simultaneously optimizes the schedule efficiency measured with the original objective \(\sum_{j=1}^{n^{\prime}}w_{j}W_{j}\) and the schedule stability measured with the total weighted completion times deviations, as described hereafter. _Stability criterion_: Let \(Co_{j}\) be the completion time of job \(j\), calculated when this job is scheduled for the first time. When a disruption occurs, the schedule will modify. Then, the job \(j\) will finish at a new completion time \(C_{j}\). Thus, the difference between \(Co_{j}\) and \(C_{j}\) is used to assess the instability cost. In addition, it is associated a weight \(w_{j}\) for each job \(j\), which makes it difficult to deviate the important orders. Thus, the stability measure is calculated as \(\sum_{j=1}^{n^{\prime}-z}w_{j}(C_{j}-Co_{j})\). _TWCTD_ is referred to as the total weighted completion times deviation. In this approach, the instability penalty concerns only the jobs existing in the previous schedule (\(n^{\prime}\)-\(z\)). We denoted by \(z\) the number of new arriving jobs, which are not concerned by the stability measure. However, once the new job is scheduled in its first sequence, it will be concerned by the stability criterion in the ulterior schedules. When \(w_{j}(C_{j}-Co_{j})>0\), a deviation has produced. Then, an instability penalty is calculated. Moreover, both objectives are associated with \(\alpha\) the efficiency-stability coefficient. It has been introduced for helping the decision maker to find a trade-off between the two optimized aspects. Thus, we have associated \(\alpha\) to the efficiency criterion and \((1-\alpha)\) to the stability criterion. It is assumed that \(\alpha\) is a real number between \(0\) and \(1\). Finally, we defined the new objective function which simultaneously measures the schedule efficiency and the schedule stability as \(\alpha\sum_{j=1}^{n^{\prime}}w_{j}W_{j}+(1-\alpha)\sum_{j=1}^{n^{\prime}-z}w_{ j}(C_{j}-Co_{j})\). Thus, the dynamic rescheduling problem can be represented by: \(1|r_{i}|\alpha\sum w_{i}W_{i}+(1-\alpha)\sum w_{i}(C_{i}-Co_{i})\) in standard notation.