# Resilient Scheduling as a Response to Uncertainty in Construction Projects
## 3 Application of Resilient Scheduling on a Test Problem Instance  
### Solving the Multi-Objective Optimization Problem  
The multi-objective optimization model was applied to the test case adjusted from Reference [30] to provide a detailed description of the solving process for the multi-objective optimization problem on a single problem instance. The activity list for the problem instance is given in Table 5. The project network is shown in Figure 3, along with activities' durations in months, their resource requirement per activity, and activities' direct costs in thousand of financing units. Only one resource type is required during project execution, of which 7 units are available at any  
In the test case, we assume recognized uncertainty sources as follows:  
* Unfavorable weather conditions might interfere with the execution of Activity 2, which is performed on the construction site;
* Architectural innovation involves an element of uncertainty while implementing the Activity 4;
* Change orders in Activity 6 might cause technical difficulty for accomplishing the execution in the estimated time frame. Having in mind optimization problems with multiple objectives, a single optimal solution cannot always be found in an unambiguous manner. For example, one baseline schedule may provide a shorter makespan, with lower values considering resilience surrogate measure and profit, while another baseline schedule may have a longer duration and significantly higher values of surrogate measure and profit. Therefore, from the mathematical point of view, one optimal solution generally is not guaranteed when solving optimization problems with more than one objective function. To date, different methods have been introduced to solve the MOO problems. One of them is to combine or weigh different objective functions [67], so the range of solutions can be found as Pareto optimal points. Following this approach, it is hard to obtain a single optimal schedule, since neither of the Pareto solutions can be improved without compromising other objective values [68]. As can be seen in Reference [69], although Pareto solutions can be considered broadly comparable, it is not possible to select only one dominant solution. Another problem then arises, as it is not always clear how to combine objective functions with different measurement units. For these reasons, a hierarchical approach was used to proceed with the solving process in the current study. Since the initial assumption was that the decision-maker's preferences in obtaining the final solution were known in advance, a hierarchical approach is employed to find a single baseline solution that can be implemented for a construction project. Priorities for all objectives are assigned in decreasing order of importance as follows: timeline minimization, surrogate measure maximization, and profit maximization. This way, project duration is optimized on the uppermost level of the solving process, and the following preference is to find the highest surrogate measure value amongst all solutions that satisfy the first hierarchical objective. At the final step of solving process, a profit value is maximized for the baseline schedule, without causing degradation in earlier objective functions. The test problem was solved to optimality with an exact algorithm written in the Mathematica programming language. The overall process led to obtaining a single baseline schedule, with a makespan duration of 18 months, SM value of 0.487, and deterministic profit value evaluating to 904.12 thousand of financial units. This way, the stable baseline schedule, which is considered to be resilient under the impact of uncertainty, was produced as a result of the optimization process.