We study a primitive vehicle routing-type problem in which a fleet of $n$unit speed robots start from a point within a non-obtuse triangle $\Delta$, where $n \in \{1,2,3\}$. The goal is to design robots' trajectories so as to visit all edges of the triangle with the smallest visitation time makespan. We begin our study by introducing a framework for subdividing $\Delta$into regions with respect to the type of optimal trajectory that each point $P$ admits, pertaining to the order that edges are visited and to how the cost of the minimum makespan $R_n(P)$ is determined, for $n\in \{1,2,3\}$. These subdivisions are the starting points for our main result, which is to study makespan trade-offs with respect to the size of the fleet. In particular, we define $ R_{n,m} (\Delta)= \max_{P \in \Delta} R_n(P)/R_m(P)$, and we prove that, over all non-obtuse triangles $\Delta$: (i) $R_{1,3}(\Delta)$ ranges from $\sqrt{10}$ to $4$, (ii) $R_{2,3}(\Delta)$ ranges from $\sqrt{2}$ to $2$, and (iii) $R_{1,2}(\Delta)$ ranges from $5/2$ to $3$. In every case, we pinpoint the starting points within every triangle $\Delta$ that maximize $R_{n,m} (\Delta)$, as well as we identify the triangles that determine all $\inf_\Delta R_{n,m}(\Delta)$ and $\sup_\Delta R_{n,m}(\Delta)$ over the set of non-obtuse triangles.

翻译：我们研究一个基础的车辆路径类问题，其中$n$个单位速度的机器人从一个非钝角三角形$\Delta$内部的一点出发，其中$n \in \{1,2,3\}$。目标是设计机器人的轨迹，使得访问三角形所有边的总访问时间（最短完工时间）最小。我们首先引入一个框架，将$\Delta$细分为不同区域，每个区域内的点$P$对应特定类型的最优轨迹，涉及边的访问顺序以及最小完工时间成本$R_n(P)$的确定方式（$n\in \{1,2,3\}$）。这些细分是我们主要结果的起点，即研究不同车队规模下的完工时间权衡。具体而言，我们定义$ R_{n,m} (\Delta)= \max_{P \in \Delta} R_n(P)/R_m(P)$，并证明在所有非钝角三角形$\Delta$上：(i) $R_{1,3}(\Delta)$的取值范围为$\sqrt{10}$到$4$，(ii) $R_{2,3}(\Delta)$的取值范围为$\sqrt{2}$到$2$，(iii) $R_{1,2}(\Delta)$的取值范围为$5/2$到$3$。在每种情况下，我们精确确定了每个三角形$\Delta$中使$R_{n,m} (\Delta)$最大化的起始点位置，并识别出在非钝角三角形集合上决定所有$\inf_\Delta R_{n,m}(\Delta)$和$\sup_\Delta R_{n,m}(\Delta)$的三角形。