**5. Discussion**

The purpose of this study was to adapt *MM*AS to the QTSP-PIC and compare its performance with the ACO variants proposed in [5]. As expected, MS-*MM*AS proved to be competitive regarding the other ACO variants proposed to solve QTSP-PIC. Similarities and differences that were observed in the results are discussed in section 5.1. The limitations of the study are discussed in Section 5.2.

#### **5.1 Comparison between ACO algorithms**

The ACO algorithms proposed by Silva et al. [5] showed to be a viable method for solving QTSP-PIC. Yet, the performance of AS and ACS algorithms, when compared to MS-*MM*AS, was rather poor for the benchmark set studied. MS-*MM*AS improved the results achieved by AS in 134 instances. Compared to ACS, MS-*MM*AS performed better in 136 instances. The results achieved by MS-*MM*AS improved those produced by MS-ACS in 93 instances. It is interesting to note that the MS-ACS algorithm performed slightly better on the large instances than the MS-*MM*AS. The Friedman test and Nemenyi post-hoc ranked these two ACO algorithms with the same scale for the most instance groups, which means that difference between the results achieved by the MS-ACS and MS-*MM*AS was significantly small.

These observations are also supported by the variability results of each ACO algorithm. Metric *ν* showed that MS-*MM*AS was the algorithm that achieved the best know solutions of the benchmark set in most cases. Metric Φ showed that the MS-*MM*AS performed slightly better overall on the benchmark set than the MS-ACS. A possible explanation for this is that the MS-ACS variation might converge to a local minimum faster than the MS-*MM*AS.

Results presented in this study showed that the MS-*MM*AS algorithm is better suited than the other three ACO variants proposed in [5] to solve QTSP-PIC. This suggests a positive impact of the implementation design proposed in this study and a contribution to the *MAX-MIN* Ant System state of the art.

#### **5.2 Limitations**

Due to limited time, parallel computing techniques could not be tested to improve the performance of MS-*MM*AS. A previous study done by Skinderowicz [10] investigated the potential effectiveness of a GPU-based parallel *MAX-MIN* Ant System in solving the TSP. In this study, the most promising *MM*AS variant was able to generate over 1 million candidate solutions per second when solving a large instance of the TSP benchmark set. Other techniques can improve the MS-*MM*AS design and could not be tested due to the lack of time:

