# Project Robust Scheduling Based on the Scattered Buffer Technology
## 4 Unit activity Slack Algorithm (UAS)
### Metric of the Solution Robustness
#### Three Types of Delay in the Solution Activity  
In the resource flow network, because of the constraining relation among activities, the delay of predecessors will transfer in its path; therefore, every solution activity will be affected by the delay of its predecessors. According to the different buffer size among the predecessors and the solution activity, and whether the buffer factors exists, the influence of delay of any predecessor on the solution activity can be divided into the following three types. (1) There is buffer time among the solution activity and its immediate predecessor. The characteristic of this type of delay is that there is a buffer time among the solution activity and its immediate predecessor activities, and there may be time buffers among the predecessors. Thus, when the schedule is executed, if a predecessor delays, this delay effect is transferred through its longest path to other activities. As a buffer exists between the solution activity and its immediate predecessor, it will absorb or resist this disruption. It is good for the stability of the schedule. It makes the start of the solution activity less likely to delay. For example, in Figure 1a,b, respectively, is the resource flow network and the scheduling. The start time vector of each activity in the scheduling is \(st=[0,9,0,0,9,17,6,25,15,23,31,36]\). Taking the solution Activity 11 as the analysis object, this paper investigates the delay effect of predecessor Activity 2 on the solution Activity \(11\) (all predecessors between the predecessor Activity 2 and the solution Activity 11 are shown in the red region in Figure 1a). In the figure, the longest path between the predecessor Activity 2 and its solution activity is \(2-6-8-11\); in this longest path, there is two days (\(31-(25+4)=2\)) of buffer time between the solution Activity 11 and its immediate predecessor Activity 8, this buffer time eases the delay effect of predecessors on the solution activity to a certain extent and guarantees the stability of the schedule. Obviously, more buffer times in this type are more favorable to the stability of the schedule. (2) There is no buffer time between the solution activity and its immediate predecessor and among the predecessors. The characteristic of this type of delay is that, on the longest path between the predecessors and its solution activity, there is no buffer time between the solution activity and its predecessors. Any delay of the predecessors will be totally transferred to the solution activity through the longest path, which makes the solution activity delayed. Thus, in this type of delay, the delay effect of predecessors on the solution activity is the most one, and, the greater the number of predecessors, the bigger the risk is. This requires that any predecessors must be executed according to the schedule when the schedule is executed, otherwise the start time of the solution activity might be postponed. In the above example, as shown in Figure 2a,b, this paper investigates the delay effect of the predecessor Activity 4 on the solution Activity 11 (all the predecessors between the predecessor Activity 4 and the solution Activity 11 are shown in the red region in Figure 2a). In all paths between the predecessor Activity 4 and the solution Activity 11, the longest path is \(4-7-5-9-10-11\). In this path, no buffer time between the solution Activity 11 and its immediate predecessor or among the predecessors exists. Therefore, any delay of any predecessor would make the solution Activity 11 postponed. Figure 1: The impact analysis of the delay of predecessor Activity 2 on the solution activity: (**a**) resource flow network and (**b**) minimal makespan schedule. (3) There is no buffer time between the solution activity and its immediate predecessor, while there are buffer times among the predecessors.