## 6 Recent Applications of Fuzzy Simheuristics
Notice that the lowest costs (both the individual ones for each instance as well as the average ones across instances) are achieved for the deterministic scenario. Here, a standard metaheuristic algorithm can provide efficient solutions, OBD-D, with an average cost of 862.7. As usual, the average and individual costs increase with the level of uncertainty. Hence, when a simheuristic is used in the stochastic scenario, the best solutions (OB-S) show an average cost of 870.8). Likewise, when considering a stochastic-fuzzy scenario, a fuzzy simheuristic provides solutions OB-SF with an average value of 872.6. Then, when considering a pure fuzzy scenario, a fuzzy approach generates solutions OB-F with an average value of 874.6. In addition, [PERSON] et al. [45] proposed an extended simheuristic algorithm with a fuzzy component to solve the permutation flow shop problem under both stochastic and fuzzy uncertainty. The proposed approach modeled fuzzy processing times based on job and machine type and used an FIS with triangular fuzzy membership functions to determine the processing time. The paper also highlighted the importance of considering statistical metrics beyond the expected makespan and showed how reliability concepts can provide probabilistic information about the proposed solutions. The results obtained  
Figure 5: Results for the VRP using a fuzzy simheuristic. when solving a set of classical instances are summarized in the box-plots displayed in Figure 7. In the PFSP, the goal is to minimize the makespan. Notice that the lowest makespans (both the individual ones for each instance as well as the average ones across instances) are achieved for the deterministic scenario. Here, a standard metaheuristic algorithm can provide efficient solutions, OBD-D, with an average makespan of 6317.7. When the former scheduling plans generated by the metaheuristic are applied in a stochastic scenario, solutions OBD-S, their performance is worse (average reward of 6379.7) than the best solutions provided by a simheuristic algorithm (OB-S, with an average makespan of 6371.1). As expected, the average and individual makespans increase with the level of uncertainty. This can be observed when considering a stochastic-fuzzy scenario, where a fuzzy simheuristic provides solutions OB-SF with an average value of 7325.2, which is slightly better than the average makespan provided by the deterministic solution when it is applied to a fuzzy-stochastic environment (OBD-FS, with a value of 7400.01). Also, a similar effect can be observed when considering a pure fuzzy scenario, where a fuzzy approach generates solutions OB-F with an average value of 7991.6, which again is slightly better than the average makespan provided by the deterministic solutions when employed in the fuzzy scenario (OBD-F, with a value of 8057.0). Finally, [PERSON] et al. [46] introduced a fuzzy simheuristic for solving the time-capacitated arc routing problem under both stochastic and fuzzy uncertainty. In the proposed approach, the results obtained when solving a set of classical instances are summarized in the box-plots displayed in Figure 8. In the TCARP, the goal is to minimize the total cost incurred by the fleet of vehicles during their trips. As always, the lowest costs (both the individual ones for each instance as well as the average ones across instances) are achieved for the deterministic scenario--this scenario constitutes an ideal one in which we have perfect information and no uncertainty. Here, a standard metaheuristic algorithm can provide efficient solutions, OBD-D, with an average cost of 144.8. When the former routing plans generated by the metaheuristic are applied in a stochastic scenario, solutions OBD-S, their performance is slightly worse (average reward of 153.9) than the best solutions provided by a simheuristic algorithm (OB-S, with an average cost of 153.0). As expected, the average and individual costs increase with the level of uncertainty.