## 3 Proposed Heterogeneous Algorithm for Efficient-Path Detection
### Heterogeneous ACO-PT Algorithm for Vehicle Management
The local trail update \(Lt_{u}\) can be determined as follows:  
\[Lt_{u}=(1-e_{s})P_{A(\epsilon 6)}+(e_{s})P_{A(0)} \tag{4}\]  
where \(e_{s}\) is the speed of evaporation and \(P_{A(0)}\) is the initial pheromone value that is allocated to all paths. Following the construction of a viable route by a set number of vehicles, a global trail update \(G_{tr}\) is carried out by adding pheromones to each link in the efficient route discovered by a leading vehicle. The global trail update \(G_{tr}\) can be calculated as follows:  
\[G_{tr}=\Big{(}(1-e_{s})P_{A(\epsilon 6)}\Big{)}G_{tr}+(e_{s})*(r_{o})^{-1} \tag{5}\]  
where \(r_{o}\) is the overall distance. This update enhances the likelihood that future routes will use the pathways found in the best solutions and supports the usage of shorter best routes. When necessary, PT finds new paths. When a vehicle lacks a valid routing-table entry and must send certain events or data to the main office, it creates a forward message and broadcasts it to all nearby vehicles. The local routing database of an intermediary vehicle that has received this forward message is searched for a viable route to the specified destination. If the search is successful, the receiving vehicle creates a backward data packet,which is then delivered through the reverse connections as a unicast message back to the source vehicle from which the initial request was made. The vehicle establishes a reverse link to the node from which the forward message is received and broadcasts the forward data packet if it has no viable path to the destination. The destination node creates a backward data packet that is also broadcasted back to the source vehicle when it obtains the forward data packet. Each intermediary vehicle adjusts its routing table to build up a forward pointer upon receiving the backward data packet and then sends the backward message to its next hop by applying the reverse pointer. The procedure continues until the first source vehicle receives the reverse message. When a packet reaches a vehicle, the pheromone for the source of the data packet is incremented by the reward \(R_{e}\). The reward has a single nominal value. Only data packets sent to the vehicle are processed. If the vehicle is the recipient of the data packet at the next hop, this situation is addressed in advance. The pheromone-update process \(P_{u}\) requires determining when a data packet is received at the source \(Sh\) vehicle from the previous hop \(Ph\). \[P_{u}(Ph,Sh)=(P_{u}(Ph,Sh))+R_{e} \tag{6}\]  
The reward can be determined as follows:  
\[R_{e}=\frac{V_{tn}}{E_{in}-(E_{mi}-V_{n})/(E_{av}-V_{n})} \tag{7}\]  
where \(V_{tn}\) is the total number of vehicles, \(E_{in}\) is the initial energy of the vehicles, \(E_{ini}\) is the minimum energy of the path traversed by the forward lead vehicle, and \(E_{av}\) is the average energy of the path traversed by the forward lead vehicle. The nominal pheromone-evaporation interval is set to one second for the vehicle, which is known as the decay time, and it represents pheromone degradation, which is determined as follows:  
\[T_{ev}(\dot{n},\mathcal{A})=T_{ev}\Big{(}\dot{n},\mathcal{A}^{cf^{-v}}\Big{)} \tag{8}\]  
where \(T_{ev}\) is entry into the PT for the vehicle, \(\dot{n}\) is the neighbor index for the vehicle, \(\mathcal{A}\) is the destination index for the vehicle, and \(ef^{-v}\) is the evaporation factor. Some vehicles need a slow rate of decay. To account for pheromone loss \((1-\mathcal{G})\), the percentage of the original value is regularly deducted from each value in the pheromone table, which can be illustrated as follows:  
\[T_{ev}(\dot{n},\mathcal{A})=(1-\mathcal{G})T_{ev}(\dot{n},\mathcal{A}) \tag{9}\]  
where \(k_{1}\) is the distance of the vehicle from the source in the horizonal axis and \(k_{2}\) is the distance of the vehicle from the source in the vertical axis.