## 4 The Proposed Approach
### Methods and Algorithms  
After formulating the problem at hand, we now show how the ED_Foraging approach works and explain the details of the proposed DTA method, called Hotspot DTA. As illustrated in Figure 1, the approach starts by initializing the state of all resources to susceptible and randomly distributing them around the arena. Then, it randomly changes some of the resources' states from susceptible to exposed via the DD_Model. Finally, robots are dispatched from the collection zones to follow randomly selected paths, and the constraints are updated online at each time. To allow a swarm of robots to work together and collect resources without prior knowledge about their distributions or deadlines, the hotspot DTA method is proposed using the _Robot_Search_and_Rescue_ function, which is continuously executed, as shown in Figure 2a. In this function, two important actions are configured. The first action is that when the robot finds a resource, it picks it up and delivers it to the closest collection zone. While the robot is picking up the resource, the robot scans its surroundings to compute its spot intensity \(H_{t}\) and local density \(D_{t}\). As described in (7) and (8), \(D_{t}\) is measured to allow the robot to remember the location of a previously found target and return to the same location if it is dense, while \(H_{t}\) is computed based on the average expiration time of the expected expired deadlines surrounding the robot to prioritize more critical locations. Figure 1: ED_Foraging “Main function”. The second action is that when the robot reaches a collection zone, it delivers the resource that it is carrying. In this case, the robot reports the collected information (\(H_{t}\)) to the collection zone, which is shared with the other robots. Then, the robot checks the density \(D_{t}\) using the Poisson CDF, as discussed in the previous section. If the Poisson CDF \(<\rho\in(0;1)\), then the last visited site is chosen, and the robot returns to a pre-visited region with a high density of resources. However, if the Poisson CDF condition is not satisfied, the robot checks whether there are any hotspots around it. Using the _HotspotExist_ function, the robot scans the hotspot list stored in the current collection zone to choose intense regions based on \(H_{t}\) (Figure 2). The chosen hotspot is then considered as the navigation target so that the robot goes to and searches for resources in that area. Note that the spot intensity (weight) is initialized by \(H_{t}\) via (8) and updated continuously by all the robots using the _Update_Hotspot_Intensity_ function; this function will be discussed later in this section. Going back to Figure 2a, if there is no intense hotspot or dense region found, the robot randomly searches any region. During the random walk, if the searching time exceeds the search probability, the robot gives up searching that region and randomly searches in another region. While the algorithm is executing the _Robot_Search_and_Rescue_ function, it also updates the DD_Model and the deadlines accordingly. From Figure 3a, the _Update_DD_Model_ function is used to continuously check the constraints and update the resource state. Figure 2: ED_Foraging “Hotspot DTA functions”. (**a**) Important actions performed by the robot during its search; (**b**) Assigning a robot to a hotspot using its weight, initialized as \(H_{t}\).