5.3.3. Scalability

We compared the scalability of the three SC-based MAPP solvers in terms of planning for a large system size (*m* > 50). Figure 9 presents the success rate, average additional cost (i.e., how much more cost than the individually optimal path), and run-time ratio over SC-CBS under different thresholds *T*, where the run-time ratio of SC-CBS is compared to itself and thus is constant. SC-A\* has the slackest constraint (*T* = 0.35, *δ* = 9.0) but poorest performance because of the prohibitively large search space. SC-CBS has the best success rate because of the property of the decoupled searching. However, this is at the expense of path cost. SC-M\* performs decently in terms of both the success rate (significantly superior to SC-A\*) and cost (noticeably lower than SC-CBS) as the number of agents increases.

**Figure 9.** Success rate, cost, and run time ratio of the three SC-based MAPP solvers under different *T*.

The run time of the SC-M\* is generally longer than that of SC-CBS. In another experiment, we observe that the run-time ratio of SC-M\* over that of the SC-CBS starts to decrease after a peak. This is because we force all algorithms to terminate after 1000 s, and both curves will converge to value one when their success rates decline to zero. We conducted another scalability experiment with different offsets *δ* (given *T* = 0.25) and observe the same results in terms of scalability. Figure 10 shows the experimental results.

**Figure 10.** Success rate, cost, and run-time ratio of the three SC-based MAPP solvers under different *δ*.

Considering the scalability and path cost altogether, SC-M\* demonstrates its overall advantages over alternative SC-based solvers.

#### **6. Conclusions**

This paper proposes SC-M\*, a generalized version of M\* with soft-collision constraints on common resources, which can scale to solving the multi-agent path planning problem in the soft-collision context. The SC-M\* tracks the collision score of each agent and place agents, whose collision scores exceed some thresholds into a soft-collision set for sub-dimensional expansion. We show that the SC-M\* has advanced flexibility and scalability for efficiently solving MAPP problems in the soft-collision context and can handle complex environments (e.g., with multiple types of agents requesting multiple types of resources). We compare the SC-M\* to other SC-based MAPP solvers and show the advantages and trade-offs of the SC-M\* against baselines in terms of path cost, success rate, and run time.

Future work will focus on leveraging advanced variants of M\*, such as EPErM\*, ODrM\*, etc., to remove the basic A\* component in our planner. We believe that better performance can be obtained this way because these variants improve the coupled planner and policy generator (two important components in the basic M\*), which are directly related to the M\* bottlenecks that limit the planning scalability. We are also interested in applying SC-M\* to real-world applications for case studies. One promising research direction is to use the proposed algorithm to serve the passengers in public transits. It is expected that SC-M\* will handle large-scale mobility demands in cities

**Author Contributions:** Conceptualization, R.S., P.S. and M.M.V.; Methodology, R.S. and M.M.V.; experimental design, R.S. and P.S.; software, R.S.; analysis, R.S.; writing, original draft preparation, R.S.; writing, review and editing, R.S., P.S. and M.M.V.; supervision, M.M.V. and P.S.; and funding acquisition, M.M.V. and P.S.

**Funding:** This research was funded by the Fundação para a Ciência e a Tecnologia (FCT), the Portuguese national funding agency, under the Sensing and Serving a Moving City (S2MovingCity) project (Grant CMUP-ERI/TIC/0010/2014).

**Acknowledgments:** The authors would like to thank Stephen F. Smith and Carlee Joe-Wong for helpful discussions.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:

