A Heuristic for Dual Mode Routing with Vehicle and Drone

In this paper we consider the problem of finding the triplet (S,${\pi}$,f), where $S{\subseteq}V$, ${\pi}$ is a sequence of nodes in S and $f:V{\backslash}S{\rightarrow}S$ for a given complete graph G=(V,E). In particular, there exist two costs, $c^V_{uv}$ and $c^D_{uv}$ for $(u,v){\in}E$, and the cost of triplet (S,${\pi}$,f) is defined as $\sum_{i=1}^{{\mid}S{\mid}}c^V_{{\pi}(i){\pi}(i+1)}+2$ ${\sum_{u{\in}V{\backslash}S}c^D_{uf(u)}$. This problem is motivated by the integrated routing of the vehicle and drone for urban delivery services. Since a well-known NP-complete TSP (Traveling Salesman Problem) is a special case of our problem, we cannot expect to have any polynomial-time algorithm unless P=NP. Furthermore, for practical purposes, we may not rely on time-exhaustive enumeration method such as branch-and-bound and branch-and-cut. This paper suggests the simple heuristic which is motivated by the MST (minimum spanning tree)-based approximation algorithm and neighborhood search heuristic for TSP.