## 3 The Algorithm
### SaNSDE Algorithm

Table 3: The information of processing time of bids in \(B\). borhood search-based DE (NSDE) algorithm [49] were used for comparison with the proposed algorithm. There are two reasons to compare it with these algorithms. First, these algorithms had been applied to the special case of the problem considered in this paper, in which the energy consumption factor is not considered in the objective function (i.e., \(w_{2}=0\) in (6)) but the constraints are the same. Second, these algorithms had also been successfully applied to solve complex constrained optimization problems with binary decision variables such as the ones in the studies [50; 51]. Therefore, we compared the proposed algorithm with these eight algorithms. We performed experiments by applying these algorithms to solve ten instances. We recorded and analyzed the results to compare the performance and efficiency of the different algorithms. All the algorithms mentioned above are population-based algorithms. Population size (\(NP\)) is a parameter for all these algorithms. As the performance of different evolutionary algorithms may depend on population size, we consider a small population size of 10, a moderate population size of 30 and a larger population size of 50 to conduct the experiments in this study. The small population size is used to test whether all these algorithms can work effectively even if the total number of individuals in the population is small. By comparing the results obtained with a small population size of 10, a moderate population size of 30 and a larger population size of 50, we will be able to know which algorithms are sensitive to the population size parameter. In this subsection, we first present the results of experiments for the population size \(NP=10\). The results of experiments for the population sizes \(NP=30\) and 50 will be presented next. In addition to the population size parameter, the other parameters used in the different algorithms are as follows. The parameters used in the discrete NSDE algorithm are as follows. \[V_{\max}=4\]

\[CR=0.5\]

\(F_{i}=0.5r_{1}+0.5\), where \(r_{1}\) is a random value with a Gaussian distribution \(N(0,1)\). The number of iterations: 10,000

The parameters used in the discrete Particle Swarm Optimization algorithm are as follows. \[V_{\max}=4\]

\[c_{1}=0.4\]

\[c_{2}=0.6\]

\[w=0.4\]

The number of iterations: 10,000

The parameters used in the discrete Differential Evolution algorithm are as follows. \[V_{\max}=4\]

\[CR=0.5\]

\(F_{i}\): a value arbitrarily selected from uniform (0, 2)

The number of iterations: 10,000

Tables 5 and 6 show the results for the population size \(NP=10\), obtained by applying the SaNSDE algorithm, a variant of the neighborhood search-based DE (NSDE) algorithm, the PSO algorithm and six standard DE algorithms. In terms of performance, the results of Table 5 indicate that the SaNSDE algorithm either outperforms or performs as well as the NSDE algorithm and PSO algorithm for most instances, with the exception of Case 4. The average fitness function values in Tables 5 and 6 are shown in the bar charts of Figures 5 and 6, respectively. The results of Table 6 indicate that the SaNSDE algorithm either outperforms or performs as well as the six standard DE algorithms in most instances, with the exceptions of Case 4, Case 5 and Case 6. The number of best solutions found by the SaNSDE algorithm, the NSDE algorithm, PSO algorithm, DE1 algorithm, DE2 algorithm, DE3 algorithm, DE4 algorithm, DE5 algorithm and DE6 algorithm are 7, 1, 2, 2, 1, 2, 0, 1 and 1, respectively. Therefore, in terms of the number of best solutions found, the SaNSDE algorithm outperforms the other algorithms.