## 4 Results  
In this section, the methods are analyzed based on their computing-time performances. We focus on this metric because our results have not shown significant differences among the methods for other metrics, e.g., optimality gap and upper bounds. In addition to analyzing averages of the metric, we use the well-known performance profile [34]. Multimedia files containing the main results for the set \(\mathcal{E}\) are attached to this work. Figure 1 presents the performance profiles of the methods considering the experiments \(\mathcal{E}\). In Figure 1, \(\rho(\tau)\) and \(\tau\) are, respectively, the probability that the performance ratio of a given method is within a factor \(\tau\) of the best ratio, as in Reference [34]. Applying the classical Benders decomposition  
(BD) on the set {40, 50, 70} \(\times\) {A} \(\times\) {L, M, H} \(\times\) {Combination 1} results in the convergence only of the problem in \([70]\times\{\)A} \(\times\) {M} \(\times\) {Combination 1}, for which BD converges to a 1%-optimal solution in 1281.42 s. Thus, it is reasonable to expect that the classical BD would also perform poorly for the remaining experiments \(\mathcal{E}\). In Figure 1, we see that the dynamic asynchronous LD outperforms all other methods for most instances \(\mathcal{E}\). Its performance ratio is within a factor of 2 from the best ratio for about 500 instances (about 92% of the total). Moreover, the static asynchronous LD has a reasonable overall performance--it is within a factor of 2 from the best ratio for more than 400 instances. Moreover, we see that the dynamic-allocation strategy provides significant improvements for both the asynchronous and synchronous LD approaches. The dynamic synchronous LD converges faster than its static counterpart for most of the experiments. Figures 2 and 3 show the performance profiles considering only instances in \(\mathcal{E}\) with machine Combinations 1 and 2, respectively. Figure 1: Performance profiles over the set \(\mathcal{E}\). Figure 2: Performance profiles for the instances with machine Combination 1. Figure 2 illustrates that, for a distributed setting in which workers are deployed on machines with identical characteristics, the performances of the methods with dynamic allocation and those with static allocation are similar. Nonetheless, we see that the asynchronous methods still outperform the synchronous LD for most experiments. In contrast to Figure 2, Figure 3 shows that the dynamic-allocation strategy provides significant time savings for the instances in \(\mathcal{E}\) with Machine Combination 2. This is due to the great imbalance between the different machines in Combination 2--machines with processors Xeon E5-2660 v3 are much faster than those with processors Xeon X5690. Table 1 gives the average wall-clock computing times over subsets of \(\mathcal{E}\). From this table, we see that the relative average speed-up of the dynamic and static asynchronous LD over the entire set \(\mathcal{E}\) w.r.t. The static synchronous LD are 54% and 29%, respectively--considering the dynamic synchronous LD, the speed-ups are 45% and 16%, respectively. Moreover, we see that the time savings are more significant for harder-to-solve instances, e.g., instances with high load and/or low-quality initial stability centers. Additionally, Table 1 shows that the dynamic asynchronous LD provides considerable reductions in the standard deviations of the elapsed computing times, in comparison with the other methods. For example, for the problems with high load level (H), the dynamic asynchronous LD has a standard deviation of about 16%, 13%, and 27% smaller than that of the static asynchronous LD, dynamic synchronous LD, and static synchronous LD, respectively. Based on the data from Table 1, we can compute the speed-up provided by our proposed dynamic ASYN LD w.r.t., and the other three variants are considered here.