## 2 Literature Review
### Measuring Uncertainty  
Given that project execution is always subject to some degree of uncertainty, much research has been dedicated to the generation of buffer times to handle eventualities. All of these models present different advantages, but nearly all present a major drawback in that they are subject to the extent of the scheduler's experience. One of the most widely-used approaches is fuzzy systems, which is especially well-suited to handle vague information. [PERSON] and [PERSON] [38] proposed a fuzzy critical chain method, which developed a desirable deterministic schedule under resource constraints and added a project buffer to the end of the schedule to deal with uncertainty. In an attempt to handle uncertainty in activity duration, [PERSON] et al. [39] used fuzzy set theory to represent the uncertainties of activity duration. Meanwhile, [PERSON] et al. [40] presented a bi-level model for project scheduling in a fuzzy random environment.  
Stochasticity presents a different approach to handle uncertainty. [PERSON] et al. [41] developed a technique for stochastic network resource allocation decisions with imprecise durations for each activity; [PERSON] and [PERSON] [42] used the Estimation of Distribution Algorithm (EDA), a kind of stochastic optimization algorithm based on statistical learning; and most recently, [PERSON] et al. [43] proposed a multi-objective Estimation Distribution Algorithm (moEDA) to minimize a schedule's robustness and expected makespan.  
The approach presented here uses the concept of entropy, a key term in thermodynamics, which does not depend on the scheduler's experience. In [44], [PERSON] and [PERSON] formulated a way to measure activity entropy as shown in Equation (1), with complete project entropy calculated using Equation (2). \(E_{i}\), represents the set of unfavorable events for activity \(i\). The value \(\delta t\) is a relevant time interval and is dependent on the nature of the project in the sense that its value is negatively correlated to the level of project risk. From a practitioner's point of view, we can think of \(\delta t\) as checkpoints or meetings in which the project manager receives updated project status information. The higher the risk of a schedule disruption, the more control needs to be exerted, requiring more frequent updates. For this study, we fixed the value of \(\delta t=1\) based on a sensitivity analysis performed in our previous research. The slack, \(s_{i}\), is determined by the standard forward (backward) recursion procedure. The values \(d_{ii}\), \(d_{i}\) and \(d_{il}\) respectively stand for the longest, most probable and shortestpossible durations, while \(EFT\) and \(LFT\) respectively represent the earliest and latest finish times with the corresponding durations indexed. \(E_{i}\) results as the time difference between \(LFT_{d_{i}}\) and \(EFT_{d_{in}}\), as shown in Equation (3).  
\[S_{ID}= -\frac{E_{i}}{(d_{in}-d_{il})\ln\left(\frac{\delta t}{d_{in}-d_{il}}\right)} \tag{1}\] \[S= -\sum_{ID}S_{ID}\] (2) \[E_{i}=(d_{in}-d_{i})-s_{i}= EFT_{d_{in}}-LFT_{d_{i}} \tag{3}\]  
From these equations, one can see that an activity's entropy depends solely on the range of the estimation for its duration, i.e., the larger the difference between \(d_{in}\), \(d_{i}\) and \(d_{il}\), the more uncertain the scheduler considers the activity to be, and thus, the higher the value of its entropy. Figure 1 gives a graphical representation of \(E_{i}\) and \(\delta t\). Here, the solid grey bar represents an activity scheduled with a most probable duration \(d_{i}\), and slack, \(s_{i}\), is the time interval between \(EFT_{d_{i}}\) and \(LFT_{d_{i}}\). The white bar shows the \(EFT_{d_{il}}\) and \(EFT_{d_{in}}\); \(E_{i}\) represents the unfavorable events; and the relevant time interval parameter defined by the user is shown as \(\delta t\).