# A Sim-Learnheuristic for the Team Orienteering Problem: Applications to Unmanned Aerial Vehicles
## 5 Computational Experiments  
The proposed sim-learnheuristic algorithm was implemented using Julia 1.8.2 and tested on a personal computer equipped with an Intel Core i7 processor operating at 2.8 GHz and 16 GB of RAM. The algorithm's computational time was set to a time limit of 60 s for all instances. The \(\beta\) parameter for the geometric distribution was randomly assigned from the interval \((0.1,0.3)\) after a quick tuning process over a random sample of instances which established a good performance. For the exploratory and refinement stages of the simulation phase, we conducted 100 and 1000 simulation runs, respectively. To validate the proposed methodology, we randomly selected 23 instances from a well-known benchmark presented next and conducted 10 separate executions for each instance, utilizing varying initial seeds for the algorithm. Among the results from these runs, we reported the best solutions based on the highest collected reward. Given the unavailability of publicly accessible instances for the dynamic and stochastic TOP, we decided to enhance the benchmark instances introduced by [PERSON] et al. [1] for the deterministic version of the TOP. The benchmark collection consists of 320 examples, categorized into seven distinct sets. Each example within a set is identified using the naming convention p.x.z, where x represents the set number, y denotes the number of drones, ranging from two to four depending on the specific instance, and z indicates the maximum driving range. These benchmark instances have been widely used in previous research to assess the performance of algorithms in solving the deterministic TOP [32; 33; 34]. However, for our study, we extended these instances to encompass four different scenarios: deterministic, stochastic, dynamic, and hybrid scenarios. In the deterministic scenario, the travel time for each route is predefined. The travel time is given by the Euclidean distance defined between each pair of nodes of the benchmark instances. In the stochastic scenario, we introduce uncertainty into the travel times between nodes. Each edge \((i,j)\in A\) in the directed graph \(G=(V,A)\) is now defined by a travel time, \(T_{ij}=T_{ji}>0\), which is not deterministic but follows a best-fit probability distribution function with a mean value \(E[t_{ij}]>0\). In our computational experiments, we used the Log-Normal probability distribution to model the random travel times. The Log-Normal distribution is preferred over the Normal distribution when modeling non-negative random variables. It has two parameters, namely the location parameter \(\mu\) and the scale parameter \(\sigma\). These parameters can be determined based on the properties of the Log-Normal distribution considering the stochastic travel times between nodes \(i\) and \(j\) are assumed to be as follows: \(E[T_{ij}]=t_{ij}\) (i.e., the travel costs of the deterministic instances), and \(\text{Var}[T_{ij}]=\subset t_{ij}\) for all \(i,j\in\{0,1,2,\ldots,n+1\}\). The parameter \(c\) serves as a design parameter enabling us to control the level of uncertainty. As \(c\) approaches zero, the results of the stochastic scenario are expected to converge with those obtained in the deterministic scenario. For our analysis, we have employed the value \(c=1\), which introduces a significant level of uncertainty as it uses the deterministic travel time as variance. A more comprehensive discussion on the implications of varying the parameter \(c\) can be found in [PERSON] et al. (2012). Specifically, the authors consider three different levels of uncertainty: low (\(c=0.05\)), medium (\(c=0.25\)), and large (\(c=0.75\)). In addition, the edges are categorized into two parts for the stochastic scenario following these conditions. If the node index is even, we classify the corresponding edge as stochastic. Otherwise, we classify the edge as deterministic. In the dynamic scenario, the travel times between nodes are predicted using a multiple regression model that takes into account the current dynamic conditions.