## 3 Results
### Case Study
The principle was applied that each reduction in the duration of activity increases its cost, which is related to the use of more expensive technology. During the performance of works by working groups, the impact of gaining experience (learning) by employees is taken into account. This results in a significant reduction in the duration of works of one type during the course of the project. In addition, the impact of forgetting how to perform a given work is taken into account depending on the size of the break between activities of one type. This results in a corresponding extension of the time needed to perform the next steps. It is assumed that these influences are determined by the modified hyperbolic learning model developed by [64]. It was assumed that the maximum efficiency is \(k=1.3\). The forgetting coefficient is \(\alpha=0.3\). The value of the \(r\) coefficient was assumed depending on the type of works at the level of 30, 90, 210, 45, and 30 for individual works, depending on the complexity of their execution. The value of the \(p\) coefficient was assumed at the level of \(r\), 2 \(r\), or 4 \(r\), depending on the experience of the working group. A construction company carrying out a given project has to respect the deadlines for the completion of individual units, which are imposed by the client. For subsequent units, these deadlines are as follows: 30, 50, 90, 130, 170, 200, and 240. The penalty for exceeding the imposed deadline for building completion is 1/100 of the average cost of the unit, i.e., 1.72, 1.24, 1.93, 2.08, 1.99, 1.84, or 2.32 thousand EUR for one day of delay for corresponding units. The reward (bonus) for the earlier performance of all activities in a building is 1/300 of the average cost of the facility, i.e., 0.57, 0.41, 0.64, 0.69, 0.66, 0.61, or 0.77 thousand EUR per day for corresponding units. The enterprise incurs indirect costs amounting to EUR 10,000 for each day of the project. The manager's task is to find a project implementation schedule that will minimize the adopted objective function, i.e., the total cost, assumed in accordance with the relationship (20). The number of possible schedules in a given project is equal to 7! :\(3^{5.7}\approx 2.5\cdot 10^{20}\). An initial schedule is adopted for the project, assuming the order of units according to their numbering (permutation of units \(\pi^{0}\) = (1,2,3,4,5,6,7)) and the choice of mode no. 2 for carrying out all activities. According to the initial schedule, the cost of the project is EUR 4,970.66 thousand, and its duration is 306.2 working days. The schedule is presented in Figure 3.  
LAF has a clear effect on the duration of the schedule. It affects not only the duration but also the cost of the project. A schedule for the baseline solution without LAF was presented in Figure 4. The cost of the project is EUR 5196.30 thousand, and its duration is 320.0 working days (both values are higher than the ones presented in Figure 3). Figure 3: Permutation \(\pi^{0}\)—construction project schedule considering LAF. Then, the two-stage algorithm was used in order to find the optimal solution. The algorithm was using the simulated annealing algorithm, as presented in Section 2.3. The resulting schedule (with specific decision variables) is presented below: \(\pi^{\text{SA}}\) = (2,1,6,3,5,4,7) and \(R\) = (\(R_{1}\), \(R_{2}\), \(R_{3}\), \(R_{4}\), \(R_{5}\), \(R_{6}\), \(R_{7}\)) where:  
* \(R_{1}\) = (2,2,2,2,1),
* \(R_{2}\) = (2,2,2,3,2),
* \(R_{3}\) = (2,1,2,2,2),
* \(R_{4}\) = (2,2,2,1,1),
* \(R_{5}\) = (3,3,2,2,2),
* \(R_{6}\) = (1,1,2,2,1),
* \(R_{7}\) = (1,1,3,2,1). The calculated total cost for this project is EUR 4622.84 thousand, and the duration is 295 working days. The cost of the project implementation is lower by 7.0% compared to the cost of the project implementation according to the initial schedule. The duration of the suboptimal schedule is 3.7% shorter than that of the baseline schedule. The obtained suboptimal schedule is presented in Figure 5.