# Deling with Uncertainty in the MRCPSP/Max Using Discrete Differential Evolution and Entropy
## 2 Coping with Uncertainty in Project Management
### Understanding Uncertainty  
Project characterization can directly influence a manager's potential response. As such, understanding project vulnerabilities is essential in the detection of uncertainty. Several researchers have proposed the information theory and entropy approach for this area [2; 26; 27; 30; 31; 32; 33; 34; 35; 36; 37]. For example, some authors in [26; 34; 35; 36] applied the entropy model as a measure of duration uncertainty or a priority rule for scheduling all activities; others [37] focused on calibrating entropy measures to better estimate uncertainty in activity durations. In this study, we consider the implications of entropy presented in [30], where an appropriate measure of project uncertainty, Schedule Entropy \(U\), borrowed from the theory of information in [38], is mathematically defined as in Equation (6):\[U=-\sum_{i}p_{i}\ln p_{i} \tag{6}\]  
where \(p_{i}\) indicates a discrete set of unfavorable activity probabilities, and the sum is extended over all the unfavorable sets of such a schedule. Since all activities durations have an estimated range between \(d_{il}<d<d_{iu}\), in which \"\(l\)\" denotes the \"lower bound\" and \"\(u\)\" denotes the \"upper bound\", and these are uniformly distributed in the interval \((d_{il},d_{iu})\), the determination of probabilities \(p_{i}\) can be expressed as in Equation (7):  
\[p_{i}=\frac{\Delta_{t}}{d_{iu}-d_{il}} \tag{7}\]  
Furthermore, the amount of time in which potentially disruptive events are within the estimated duration range, and hence still within the control of the project manager--and thus, the set of unfavorable events \(E_{i}\) for every activity--can be obtained from Equation (8),  
\[E_{i}=EFT_{d_{i}}+(d_{iu}-d_{i})-LFT_{d_{i}}=(d_{iu}-d_{i})-l_{i} \tag{8}\]  
where \(d_{iu}\) and \(d_{i}\) denote the longest and the most probable duration, respectively, and \(l_{i}\) refers to the float or slack of activity \(i\), implying that it can be delayed without delaying subsequent activities and project completion dates. Additionally, the terms \(EFT_{d_{iu}}\) and \(LFT_{d_{i}}\) refer to the earliest finish time with the longest duration and the latest finish time with the most probable duration, respectively. The entropy \(U\) of the project schedule can be thought of as all the individual entropies \(U_{i}\) of activity \(i\). Activity \(i\) can be potentially unfavorable or disruptive, if and only if it is critical to the CPM and its actual finish time is beyond the latest finish time. An entropy value for a single activity, i.e., \(U_{i}\), can be determined as in Equation (9):  
\[U_{i}=-\frac{E_{i}}{(d_{iu}-d_{il})\ln\left(\frac{\Delta_{t}}{d_{iu}-d_{il}} \right)} \tag{9}\]  
As such, Equation (10) shows the value for the total entropy in the project. \[U=-\sum_{i}U_{i} \tag{10}\]  
Based on these equations, it is understandable that an individual entropy value is subject to the estimation of the range for its duration, i.e., the more significant the difference in the interval \((d_{il},d_{iu})\), the greater the manager's perception of the activity's uncertainty and therefore the higher its entropy value. In addition, as shown below, the schedule entropy is directly affected by the order in which activities are scheduled. Figure 1 depicts the concept of unfavorable events \(E_{i}\) and the relevant time interval \(\Delta_{t}\). The solid gray bar symbolizes a scheduled activity with the most probable duration \(d_{i}\), whereas the solid yellow bar shows its most prolonged duration \(d_{iu}\). The slack \(l_{i}\) is the time interval between \(EFT_{d_{i}}\) and \(LFT_{d_{i}}\). The time difference between \(LFT_{d_{i}}\) and \(EFT_{d_{iu}}\) is denoted by \(E_{i}\). The parameter \(\Delta_{t}\) is determined by the decision-maker and is dependent on the nature of the project. Riskier projects require lower values for \(\Delta_{t}\), which acts as a checkpoint for the project manager to update the project's status and take control of its progress.