# Project Robust Scheduling Based on the Scattered Buffer Technology
## 4 Unit activity Slack Algorithm (UAS)  
### Metric of the Solution Robustness  
#### 4.1.1 Analysis of the Influencing Factors of the Predecessors on the Solution Activity  
For the convenience of narration and understanding, the following concepts are introduced. In the resource flow network, every activity will be affected by the uncertain duration of its predecessors (in this section, the predecessor refers to the direct or transitively immediate predecessor). To measure the solution robustness of the scheduling scheme, some activities need to be examined and to determine whether they start according to the predicated starting time. These activities are defined as the solution activities. In the schedule, except the start dummy activity 0, all activities need to be considered as solution activities. According to the definition of the solution robustness, under the uncertain duration, the stability of whether all the solution activities can start at the predicated time reflects the solution robustness of this scheduling. In the sub network formed by the interaction of the predecessor and solution activity, in all paths between the solution activity and its predecessors, the longest path refers to the path in which the total duration of all activities is the longest; there is at least one longest path. Because the longest path has more uncertain factors than other paths, the influence of the predecessors' delay on the solution activity transfers firstly through the longest path. Therefore, the influence of predecessors on the solution activity discussed in this paper is about what happens through the longest path. The comprehensive impact of predecessors on the solution activity is different, which makes the solution robustness of the schedule different. It is necessary to consider both the structural characteristics and the external environment of the schedule to analyze the impact of forward activities on the solution activity. Therefore, in summary, the factors that affect the solution activity can be divided into two types: static and dynamic. Static factors are related to the characteristics of the network structure in the schedule; they belong to the inherent factor here. On the longest path between the solution activity and its predecessors, static factors include the buffer slack and the number of predecessors in front of the solution activity. Dynamic factors are derived from environmental changes; they are the additional external factors. Here, only the random influence of external factors on the predecessors' durations is considered in this paper. The specific analysis is as follows. (1) The slack in front of the solution activity is denoted as \(Len_{i}^{j}-TotDu_{i}^{j}\). It refers to the difference (\(Len_{i}^{j}\)) between the start time of the solution activity \(j\) and its predecessor \(i\) on the longest path. This difference minus the sum (\(TotDu_{i}^{j}\)) of durations of the predecessors, and summing (\(\sum d_{i}^{j}\)) the expected duration of each predecessor, is \(TotDu_{i}^{j}\). The higher is \(Len_{i}^{j}-TotDu_{i}^{j}\), the greater the buffer among the predecessors becomes, the higher the ability to resist the delay in the solution activity, the better the solution robustness is, and, conversely, the worse the solution robustness is. It is obvious that the value of \(Len_{i}^{j}-TotDu_{i}^{j}\) is proportional to the value of the robustness index. (2) The number of predecessors of the solution activity is \(Num_{i}^{j}\). On the longest path, every predecessor of the solution activity actually represents a kind of uncertain factor: higher values of \(Num_{i}^{j}\) indicate uncertainty factors are more active, thus their influence on the solution robustness of the schedule is greater, the solution robustness is worse; on the contrary, the smaller the influence, the better the solution robustness is. The value of \(Num_{i}^{j}\) is inversely proportional to the value of the robustness index.