# Robot Routing Problem of Last-Mile Delivery in Indoor Environments
## 6 Conclusions
The experimental results indicated that the proposed heuristic could generate  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{\multirow{2}{*}{**Test Instances**}} & \multicolumn{4}{c}{**Proposed Heuristic**} & \multicolumn{2}{c}{**Gurobi**} & \multicolumn{2}{c}{**Gap (\%)**} & \multicolumn{2}{c}{**Gap (\%)**} \\ \cline{3-10}  & \multicolumn{2}{c}{**Step 1**} & \multicolumn{2}{c}{**Step 2**} & \multicolumn{2}{c}{**Between**} & \multicolumn{2}{c}{**Between**} \\ \hline \multirow{4}{*}{**Size**} & **\# of** & \multirow{2}{*}{**Objective**} & **Step 1** & \multicolumn{2}{c}{**Step 2**} & \multirow{4}{*}{**Condition**} & **Objective** & **Objective** & **Computation** & **Step 1 and** & **Step 2 and** \\  & & **Demand** & **Value (s)** & **Time (s)** & **Value (s)** & **Time (s)** & **Objective** & **Computation** & **Gurobi** & **Gurobi** \\  & & **Demand** & **Value (s)** & **Time (s)** & **Value (s)** & **Time (s)** & **Value (s)** & **Time (s)** & **Value** & **Value** \\ \hline \multirow{2}{*}{1} & (9, 16) & 2235.48 & 0.002 & 2233.76 & 10.8 & 2233.76 & 0.12 & 0.08 & 0.00 \\  & & (15, 24) & 3562.54 & 0.002 & 3437.12 & 98 & 3437.12 & 0.5 & 3.65 & 0.00 \\  & & (9, 13) & 1524.94 & 0.002 & 1524.94 & 13.6 & 1524.94 & 0.45 & 0.00 & 0.00 \\  & & (15, 20) & 3190.04 & 0.002 & 2751.04 & 81.9 & 2751.04 & 102.96 & 15.96 & 0.00 \\  & & (14, 20) & 2610.91 & 0.002 & 2297.63 & 45.5 & 2297.63 & 5.03 & 13.63 & 0.00 \\  & & (15, 24) & 3317.26 & 0.001 & 3087.84 & 78.2 & 3087.84 & 0.45 & 7.43 & 0.00 \\  & & (18, 27) & 3695.51 & 0.001 & 3318.06 & 112.7 & 3318.06 & 12.9 & 11.38 & 0.00 \\  & & (14, 22) & 2591.59 & 0.002 & 2591.59 & 23.9 & 2591.59 & 0.45 & 0.00 & 0.00 \\  & & (13, 18) & 2326.97 & 0.002 & 2267.11 & 43.8 & 2267.11 & 3.02 & 2.64 & 0.00 \\  & & (12, 20) & 2474.38 & 0.002 & 2400.38 & 20.5 & 2400.38 & 0.2 & 3.08 & 0.00 \\ \hline \multicolumn{2}{c}{**Average**} & **-** & **0.002** & **-** & **52.89** & **-** & **12.61** & **5.78** & **0.00** \\ \hline \multirow{2}{*}{11} & (13, 21) & 3171.21 & 0.002 & 3157.93 & 44.5 & 3157.93 & 4.85 & 0.42 & 0.00 \\  & & (23, 36) & 5861.68 & 0.001 & 5719.26 & 198.3 & 5676.98 & 37.71 & 3.25 & 0.74 \\  & & (16, 25) & 4066.25 & 0.002 & 3968.97 & 76.8 & 3968.97 & 0.74 & 2.45 & 0.00 \\  & & (22, 31) & 5185.35 & 0.001 & 4701.91 & 181.9 & - & 3600 & - & - \\  & & (19, 27) & 4321.66 & 0.002 & 3780.02 & 180.3 & 3780.02 & 615.7 & 14.32 & 0.00 \\  & & (23, 35) & 5075.59 & 0.002 & 5050.28 & 136.2 & 5042.15 & 241.4 & 0.66 & 0.16 \\  & & (27, 40) & 6217.14 & 0.003 & 5673.28 & 226.8 & - & 3600 & - & - \\  & & & & & & & & & & \\  & & & & & & & & & & & \\  & & & &satisfactory solutions (i.e., 0%, 0.11%, and 0.18% optimality gaps on average, respectively) within a short time in each case. Moreover, solutions were all obtained much faster than with Gurobi, a commercial optimization software, even in the large-sized problem case, which could not obtain optimal solutions in 6 out of 10 problem instances within an hour. Regarding future research directions, the solution obtained with the proposed approach can be validated via test simulations using robotics middleware frameworks (e.g., ROS 2 with Gazebo and Navigation 2). In addition, the mathematical model and heuristic can be revised to manage dynamic indoor situations such as avoiding obstacles and sudden reliability problems (e.g., delivery failure due to breakdown or battery discharge). They are important issues to resolve when implementing this study in real-world environments. Moreover, crowdedness, another uncertain factor of buildings, might be an important topic for further studies. Considering the crowdedness of hallways could upgrade our travel time calculator more realistically. Conceptualization, J.K. and H.J. ; methodology, J.K. and H.J. ; software, J.K.; validation, J.K. and H.J. ; formal analysis: J.K. and H.J. ; data curation, J.K.; writing--original draft preparation, J.K.; writing--review and editing, H.J. ; supervision, H.J. ; funding acquisition, H.J.