## 1 Introduction
### Hybrid Truck Multi-Drone Problem Description  
This study aimed to find the most suitable route with the lowest cost for the last mile of the hybrid truck multi-drone delivery problem. When defining the hybrid truck multi-drone delivery problem, a \(G\)-directional graph, as shown in Figure 1, is used to represent the warehouse and delivery points as nodes and the route to be followed as branches. One truck and \(k\) drones are used on the graph \(G=(V,\ E,\ C)\), a representative image of which is given in the figure. It is planned to find the least costly route, provided that all \(n\) nodes on this graph are visited once.  
A graph defined as \(G=(V,\ E,\ C)\);  
* \(V\) is a cluster of nodes on the graph.  
* \(E\) is a cluster of branches on the graph.  
* \(C\) is the cluster of the weights of the branches.  
There are \(n\) points on this graph to be delivered by the drone. These points are represented by \(d_{i}\); they are the nodes shown as the red square in Figure 1, and they are the elements of the cluster \(V_{d}\) given in Equation (1). At the same time, there are \(m\) points on this graph to be delivered by the truck. These points are represented by \(t_{i}\); they are the nodes shown as gray circles in the figure, and they are the elements of the cluster \(V_{t}\) given in Equation (2). Node \(t_{0}\), another node on the graph, is the starting point called the warehouse and returns to this point after all deliveries are made. This node is shown with a blue triangle in the figure, and it is an element of the cluster, as shown in Equation (3). Additionally, in Figure 1, straight lines represent truck movement, and dashed lines represent drone movement.  
\[V_{d}=\{d_{1},\ d_{2},\ d_{3},\ldots,\ d_{n}\} \tag{1}\]  
Figure 1: Illustration of the hybrid truck multi-drone delivery problem on a representative route.  
\[V_{t}=\{t_{1},\ t_{2},\ t_{3},\ldots,\ t_{m}\} \tag{2}\]  
\[V_{0}=\{t_{0}\} \tag{3}\]  
If all the nodes of this \(G\) graph are represented as members of a cluster \(V\), a cluster will be formed from the combination of the clusters given in the above three equations. Equation (4) shows this cluster of \(V\).  
\[V=V_{d}\ \cup\ V_{t}\ \cup\ V_{0} \tag{4}\]  
If we examine the branches on the \(G\) graph, we see that there are two types of branches. Dotted arrows show movement by the drone, while arrows without dots show movement by the truck. If these different types of branches are defined as the elements of the \(E_{d}\) and \(E_{t}\) clusters, respectively, then the \(E\) cluster with the branches of the \(G\) graph can be considered the combination of these two different branch sets. This situation is seen in Equation (5).  
\[E=E_{d}\ \cup\ E_{t} \tag{5}\]  
The weights of the branches on a weighted graph are called \(c\). In this problem, where the shortest route is trying to be found by using the \(G\) graph, the total cost value obtained at the end of the route should be minimized. When the graph in Figure 1 is examined, it can be seen that there are two different types of costs. The first of these is the cost of the branches, which are the members of the \(E_{d}\) cluster. Another is the cost of branches, which are members of the \(E_{t}\) cluster. The main purpose of defining two different costs in this way is to make deliveries using different vehicles on these two different types of branches. It is clear that deliveries by one truck and one drone will not cost the same, given that they both have different speeds. The \(c\) cost value can be represented as \(C=(c_{d},\ c_{t})\). Here, \(c_{d}\) and \(c_{t}\) can take different values only when they are negative and are defined on the functions in Equations (6) and (7).  
\[c_{d}:E_{d}\ \rightarrow\ \mathbb{R}_{\geq 0} \tag{6}\]  
\[c_{t}:E_{t}\ \rightarrow\ \mathbb{R}_{\geq 0} \tag{7}\]