# Project Robust Scheduling Based on the Scattered Buffer Technology
## 2 Problem Description  
The metric of the schedule robustness is divided into the solution robustness and the quality robustness. The solution robustness reflects the stability of the schedule execution, which is measured by the difference between the baseline schedule and the realized schedule. Quality robustness is measured by whether the project completion time coincides with the predicted makespan. Under the uncertain project environment, this paper focuses on maximizing the schedule solution robustness problem with uncertain activity durations. This paper mainly studies the robust scheduling problem with uncertain activity durations based on the resource flow network, which can be described as follows. A project can be represented by Activity On Node (\(AON\)), an acyclic simple directed graph, as \(G=(N,A\cup A_{R})\). \(N\) is defined as the set of activities. The active sequence of numbers is from 0 to \(n(|N|=n+1)\), considering the uncertainty of the project environment. Assuming activity duration \(i\) is uncertain, as is \(\widetilde{d_{i}}\), activity 0 and activity \(n\) are, respectively, the start dummy activity and the end dummy activity, and activity duration is 0. Among the activities, the immediate predecessor \(i\) and the immediate successor \(j\) must meet the precedence relation \(s_{i}+\widetilde{d_{i}}\leq s_{j}\), \(\forall(i,j)\in(A\cup A_{R})\). Assuming that the project uses \(K\) kinds of renewable resources, at any time \(t\), the amount \((\sum\limits_{i\in S_{t}}r_{ik})\) of the resource \(k\) that the activity  
where Equation (1) represents that the objective function is to maximize the solution robustness of the schedule. \(R(\Delta S,w)\) is the weight function about weight \(w\) and the deviation between the realized starting time and the schedule starting time of each activity. In different algorithms, the specific forms of the objective function representing robustness are different. The robust optimization objective function proposed in this paper is introduced in detail in the subsequent sections. Equation (2) indicates that immediate predecessor \(i\) and immediate successor \(j\) must meet precedence relation and the resource-constrained relation, in which \(s_{i}\) and \(s_{j}\) denote, respectively, the predicated start time of the activity \(i\) and the activity \(j\); and \(\widehat{d}_{i}\) indicates uncertain duration of the activity \(i\). Equation (3) indicates the constraints of the renewable resource availability in unit time, in which \(r_{ik}\) indicates the amount of resource \(k\) that the activity \(i\) needs, and \(a_{k}\) indicates the availability of resource \(k\). Equations (4) and (5) represent, respectively, the equilibrium condition of the resource flow in schedule. Under the uncertain activity duration, the above model is to seek the schedule with the maximum solution robustness. At the same time, the schedule must satisfy the precedence relations and the resource constraints. However, it is difficult to solve directly because of the uncertain duration in the robust scheduling optimization model mentioned above. The above model is generally solved using the heuristic algorithm; that is, the time buffer is added into the schedule to improve the robustness of the schedule. The buffer time refers to the idle time inserted in front of the start time of the activity. The role of the buffer time is to prevent the propagation of the disruptions in the execution of the schedule. The specific method is to build firstly a baseline schedule satisfying the precedence relations and the resource constraints, and then add protection to the baseline schedule, to generate a scheduling scheme with certain anti-interference performance, so that it is free from all kinds of disruptions of uncertain factors. This paper argues that the activity is the basic unit of the project schedule, and the stability of the schedule is demonstrated by the stability of the activity execution.