## 5 Testing the Proposed Algorithm
### Performance of the Proposed Algorithm and Comparison to the State-of-the Art Algorithms  
As stated before, we compared our algorithm with a global version of PSO proposed by [PERSON] and [PERSON] [8] and a CSO algorithm by [PERSON] et al. [11] along with an integer programming method presented by [PERSON] et al. [12]. The choice of these three attempts was justified in Section 5 above. It should be noted that the same fitness function was used among the above-mentioned studies, under the same parameters shown in Table 2. We ran each algorithm 30 times for each instance. In our experiments, we used a PC with i-4770 intel CPU at 3.40 GHz with 16 GB of RAM. The OS was Windows 64 bit. The programming language we used was the same used for the other soft computing algorithms, with which we compared our present work, i.e., ANSI C. We first present, in Table 4, the comparison between the global version of the PSO and the current local version of the PSO algorithm. For each instance, the best, worst, and average score is presented along with the standard deviation (STD) and the coefficient of variation (CV). The corresponding values for the execution times are given, too. The best score between the two algorithms is indicated by bold. It should be highlighted that the score presented in the tables below corresponds to the number of soft constraint violations, i.e., one cost unit is added for each teacher idle hour, for each case of undesirable class dispersion (repetition) and each case of undesirable teacher dispersion. Hence, the reader should not confuse the fitness value with the quality score. Although this differs from the fitness function utilized within the optimization for solution evaluation, the use of this score was typically adopted in corresponding literature to facilitate straightforward comparisons between solutions [11]. It is also worth mentioning that we did not perform any kind of statistical test in order to identify the statistical difference of results produced by several implementations the current algorithm is compared with. Indeed, contrary to a continuous objective function, the score value used allowed for the intuitive comparison between different approaches, as even a one-unit difference of the cost between two solutions signifies the satisfaction of one more soft constraint, i.e., it clearly signified a superior solution. In such a case, this indicates that a new lower bound was located and, thus, our goal was achieved. Finally, in terms of parameters, the results of the global PSO version refer to a swarm of 50 particles, 10,000 maximum iterations, 2.2% acceptance probability, and 500,000 steps for the first (and only in this case) refining procedure. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**File** & **Number of** & **Number of** & **Number of** & **Number of Teachers** & **Number of** & **Number of** & **Number of** \\
**No.** & **Teachers** & **Classes** & **Teaching** & **Involved to** & **Co-Teachings** & **Co-Teachings** & **Teachers with** & **Unavailability** \\  & & & **Hours** & **Co-Teachings** & **Hours** & **Unavailability** & **Hours** \\ \hline
1 & 34 & 11 & 385 & 9 & 36 & 12 & 224 \\
2 & 35 & 11 & 385 & 17 & 67 & 11 & 133 \\
3 & 19 & 6 & 210 & 0 & 0 & 8 & 133 \\
4 & 19 & 7 & 245 & 6 & 31 & 6 & 98 \\
5 & 18 & 6 & 184 & 0 & 0 & 10 & 161 \\
7 & 35 & 13 & 455 & 17 & 70 & 6 & 91 \\
8 & 11 & 5 & 150 & 0 & 0 & 0 & 0 \\
9 & 15 & 6 & 202 & 0 & 0 & 7 & 98 \\
10 & 17 & 7 & 210 & 0 & 0 & 0 & 0 \\
11 & 21 & 9 & 306 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Characteristics of the Benchmark data set. Evidently, the proposed algorithm compared favorably to the global-based PSO version in half of the cases, except for instances 5, 8, 9, 10, and 11, where the lowest scores were the same. Another interesting conclusion is that in the case of the difficult-to-solve instances, which were 7, 1, 2, 4, the difference in best scores achieved between the two algorithms was considerably large.