# Deling with Uncertainty in the MRCPSP/Max Using Discrete Differential Evolution and Entropy
## 2 Coping with Uncertainty in Project Management
### A Resilient Approach to Uncertainty  
As discussed in the previous sections, project managers or decision-makers must assess different controls and operating conditions. A final project schedule embodying trade-offs among various aspects reflects a stable state, characterized by its makespan, cost, risk, and net present value. However, in the presence of disruptive events, the control of activity progress may be broken, resulting in cost overruns, makespan tardiness, performance degradation, or even project failure. Therefore, this section focuses on establishing a desirable algorithm characteristic called resilience. The authors in [39] defined resilience as a process of aligning a set of adaptive capacities to provide a positive course of functioning and adaptation after a disturbance. Given changing project constraints, variables, and structures, decision-makers must adapt their preferences or objectives to arrive at another stable state. In this sense, more than one stable condition exists for project scheduling problems. In [40; 41], the term robustness is defined as \"the ability (of a schedule) to cope with small-time increments in some activities resulting from uncontrollable factors.\" Thus, resilience is the magnitude of the disruptive event absorbed before a schedule degrades to the threshold, i.e., the minimum acceptable level of individual activity performance [9]. Researchers have worked on developing an algorithm that measures how the schedule robustness of scheduled activities deviates from the actual occurrences through a resilient scheduling algorithm. [PERSON] et al. [7] determined the expected increase in activity duration due to resource breakdown, proposing a buffer time to prevent schedule disruptions. In [42], the scenario-based proactive robustness optimization (SBPRO) method was developed using the critical-chain project management (CCPM) method. [PERSON] and [PERSON] [8] applied the CCPM and considered only renewable resources at the cost of recruiting additional resources. [PERSON] and [PERSON] [43] sought to minimize the worst-case project duration by deciding on activity modes, resource allocations, and a schedule baseline. In [44], the authors included multiple alternative execution modes and allowed the switching of possibilities between different modes for the same activity during scheduled construction projects. These studies focused on procedures designed to build a robust schedule through the use of time or resource buffers. Their procedures reflect the controllable flexibility needed to produce an incremental solution based on the subsequent revelation of contingent events. Other researchers have developed optimization models for robustness measures (RMs). For instance, [PERSON] and [PERSON] [45] proposed different slack-based models to predict a schedule's robustness in relation to the single-mode RCPSP. The authors in [9] addressed resilience in mean-variance models having two types of ratios, i.e., the average interval to activity duration and the free slack-to-activity duration. Finally, [PERSON] et al. [10] improved resilience by maximizing free floats as the degree of perturbations absorbed rises. Their study depicted resilience through the alternative measure expressed in the objective func  
Figure 1: Graphical representation of \(E_{i}\) and \(\Delta_{t}\). tion, which maximizes the highest proportion of resource-technology free floats for the activities early in the schedule. In this study, after handling the issue of schedule uncertainty via entropy, we have aimed to achieve project resilience by extending the robustness model in [45] from the single-mode RCPSP to the multimode RCPSP/max, subjecting it to the constraints presented in [46]. This objective function indicates the relationships between each activity's precedence, resource usage, and slack in each available mode to maximize the robustness measures.