# Computing Idle Times in Fuzzy Flexible Job Shop Scheduling
## 4 Results
### Comparisons Based on Fuzzy \(\text{TCIT}\) Values  
As mentioned in Section 2.3.1, fuzzy \(\text{TCIT}\) values can be compared in terms of their expected value, their spread, \(S\), and their modal value position, \(MVP\). Figure 4 shows the average \(\mathbb{E}[\widehat{\text{TCIT}}]\) values obtained for each instance in the two types of experiments. Figure 4a corresponds to the average values obtained when evaluating the random schedules with the three models for computing \(\widehat{\text{TCIT}}\). Figure 4b depicts average values obtained when evaluating the schedules obtained with the tabu search using each of the three models. In both experiments, there is clearly a difference between the three methods for computing the fuzzy \(\text{TCIT}\): \(\widehat{\text{TCIT}}^{C}\) yields better average expected values than \(\widehat{\text{TCIT}}^{K}\), and this, in turn, yields better values than \(\widehat{\text{TCIT}}^{N}\). Furthermore, for the random solutions experiment, the variation in the expected value seems to depend only on the problem size and not on the flexibility level, as illustrated by the flat lines. This makes sense since, the larger the number of operations, the larger the number of possible gaps, and, as they are not optimised in any way, the longer the total idle time. However, for the searched-solution experiment, differences exist depending not only on the instance size but also on the flexibility level. This may be because greater flexibility provides the search algorithm with more opportunities to move operations between resources and fill inactivity gaps, thus reducing the total core idle time. It is remarkable that, here, \(\widehat{\mathrm{TCTI}}^{K}\) seems to yield better improvements when greater flexibility is introduced, coming closer to \(\widehat{\mathrm{TCTI}}^{C}\). On the other hand, the improvement of the naive approach \(\widehat{\mathrm{TCTI}}^{N}\) does not seem to be as pronounced when flexibility increases. Also, \(\widehat{\mathrm{TCTI}}^{K}\) expected values are significantly higher than \(\widehat{\mathrm{TCTI}}^{C}\). This is expected because, although \(\widehat{\mathrm{TCTI}}^{K}\) has been defined to reduce artificial uncertainty due to dependencies between operations, there are still relations that we cannot easily account for. Thus, the sum of idle times between operations, that is the sum of multiple subtractions, is going to have predictably more artificial uncertainty than the coarser approach, which only makes one subtraction, as the results demonstrate. When the spread is considered, the obtained graphs are very similar. The reason is that, in these benchmark instances, operations have similar-magnitude processing times with different values for each of the three components of the TFNs. As computing starting times according to Equation (11) are accumulative, the greater the number of already-scheduled operations, the larger the expected value of the TFN and the difference between the first and third components. Then, as the subtraction according to Equation (3) is computed by crossing the value of the lower and upper values of the two TFNs involved, it is predictable that larger expected values also mean larger spread values. Finally, Figure 5 shows the \(MVP\) values obtained in the experiments. Notice that, both in the random solutions and the searched solutions experiments, \(MVP\) is negative, meaning that the modal point is situated to the left of the support's midpoint; i.e., the TFNs are skewed to the right. The main reason for this behaviour is that, in all functions, we are removing negative values because negative times are not possible, but this trimming will only take place on the left side of TFNs. Also, notice that, in the searched-solution experiments, the absolute value of \(MVP\) is closer to \(-1\) compared to the random solutions.