# Generalization of ALMM Based Learning Method for Planning and Scheduling
## 6 Exemplary Problem  
An example of a problem that can be used as a difficult-to-control process due to the exponential complexity of the process of selecting control decisions may be a fleet of drones performing transport tasks to a network of airport destinations. Assume that the graph of a destination is a loaded undirected graph \(G=(I,J,P)\) in which the set of edges denotes the air routes \(J\) and the set of landing sites \(I\) represent the set of nodes partially connecting the air corridors described by the relation of interconnections \(P\subset(I\times I)\). This relation includes only those connections that are defined in the considered airspace structure of routes. In the considered example, it was assumed that the drones fleet used is the counterpart of a set of parallel machines, which may differ in their operating parameters. In particular, it is their cruising speed, transported weight and range. Airport tasks are characterized by due times and their deadlines. It is also assumed in the considered example that the operational time of the fleet is not subject to restrictions related to the access of airspace and the routes of individual ships can be determined independently while maintaining adequately safe air separation. The execution of the flight along a given route during the execution of the task is not subject to stoppages and individual tasks may be assigned weights, taking into account the priority of their implementation. The main task of the system is to generate the trajectory of decisions that control the process, taking into account the costs of the work of individual machines, the waiting times for assigning tasks and the total time of the execution of all tasks. The preliminary idea of the learning algorithm for this problem was given in [2]. Other examples that refer to simplified models of discrete processes can be indicated in other studies [23; 25]. The specification of individual elements of the AL model is very extensive. Below we will provide only those elements of the model that are necessary to explain the structure of the local criterion and methods of learning. The process state at any instant \(t\) is defined as a vector \(x=\left(x^{0},x^{1},x^{2},\ldots,x^{jM_{j}}\right)\), where \(M=M1\cup M2\). A coordinate \(x^{0}\) represents a set of corridors, namely routes to be traveled by drones to the moment \(t\). The other coordinates \(x^{m}\) describe the state of the \(m\)-th drone, where \(m=1,2,\ldots,|M|\). A state \(s(x,t)\) belongs to the set of not-admissible states if there is a route corridor whose job flight is not completed yet and whose due date is earlier than \(t\). The definition of \(S_{N}\) is as follows:  
\[S_{N}=\Big{\{}s=(x,t):\left(\exists j\in J,j\
otin x^{0}\right)\wedge d(j)<t \Big{\}}. \tag{15}\]  
A state \(s(x,t)\) is a goal if all the jobs have been completed, i.e. :  
\[S_{G}=\Big{\{}s=(x,t):s\
otin S_{N}\wedge\left(\forall j\in J,j\in x^{0} \right)\Big{\}}. \tag{16}\]  
A decision determines the flights that should be started at the moment \(t\), machines which are in the air, machines that should be serviced or prepared to be operational or the machines that should be waiting for weather reasons. Thus, the complex decision \(u=\left(u^{1},u^{2},\cdots u^{|M|}\right)\), where the co-ordinate \(u^{m}\) refers to the \(m\)-th drone and included all pieces of needed information. Based on the current state \(s(x,t)\) and the decision \(u\) taken in this state, the subsequent state \((x^{\prime},t^{\prime})=f(u,x,t)\) is generated by means of the transition function \(f\). In the course of trajectory generation in each state of the process, a decision is taken for which the value of the local criterion \(q(u,s)\) is the lowest. The local criterion takes into account a component connected with cost of work, and a component connected with the necessity for the trajectory to omit the states of set \(S_{N}\).