**6. Conclusions**

processing time. **Tables 6** and **7** (1) present detailed results concerning the average time required by the MS-*MM*AS. Data regarding the time consumption of the other

The purpose of this study was to adapt *MM*AS to the QTSP-PIC and compare its

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

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

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

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

• The application of the pseudo-random action choice rule proposed in [21];

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.

ACO variants can be seen in [5].

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**5.1 Comparison between ACO algorithms**

a local minimum faster than the MS-*MM*AS.

a contribution to the *MAX-MIN* Ant System state of the art.

design and could not be tested due to the lack of time:

• A rank-based pheromone updating rule [25];

• Hybridization with other meta-heuristics [26–28].

**5. Discussion**

significantly small.

**5.2 Limitations**

**90**

This work dealt with a recently proposed variant of the Traveling Salesman Problem named The Quota Traveling Salesman Problem with Passengers, Incomplete Ride, and Collection Time. In this problem, the salesman uses a flexible ride-sharing system to minimize travel costs while visiting some vertices to satisfy a pre-established quota. He must respect the budget limitations and the maximum travel time of every passenger. Each passenger can be transported directly to the desired destination or an alternate destination. The alternative destination idea suggests that when sharing a ride, pro-environmental or money-saving concerns can induce persons to agree to fulfill their needs at a similar destination. Operational constraints regarding vehicle capacity and travel time were also considered.

The Multi-Strategy *MAX-MIN* Ant System, a variant from the Ant Colony Optimization (ACO) family of algorithms, was presented. This algorithm uses the MS concept improved with roulette wheel selection and memory-based principles to avoid redundant executions of the local search algorithm. The results of MS-*MM*AS were compared with those produced by the ACO algorithms presented in [5]. To support MS-*MM*AS, the ride-matching heuristic and the local search heuristic based on multiple neighborhood operators proposed by [5] were reused.

The computational experiments reported in this study comprised one hundred forty-four instances. The experimental results show that the proposed ant algorithm variant could update the best-known solutions for this benchmark set according to the statistical results. The comparison results with three other ACO variants proposed in [5] showed that MS-*MM*AS improved the best results of MS-ACS for ninety-three instances, and a significant superiority of MS-*MM*AS over AS and ACS.

The presented work may be extended in multiple directions. First, it would be interesting to investigate if the application of the pseudo-random action choice rule [20] could improve the MS-*MM*AS results. Another further promising idea is the use of pheromone update rule based on ants ranking [25]. Extension of the MS-*MM*AS implementation design with parallel computing techniques [10] and hybridization with other meta-heuristics [26–28] is other interesting opportunity for the future research.
