# Generalization of ALMM Based Learning Method for Planning and Scheduling
## 6 Exemplary Problem
The first components constitute a sum of \(\Delta Q(u,x,t)\) and \(\Delta\widehat{Q}(u,x,t)\), where \(\Delta Q(u,x,t)\) denotes the increase of work cost as a result of realizing decision \(u\) and \(\Delta\widehat{Q}(u,x,t)\) is the lower estimation of the cost of finishing the set of headings matching the final section of the trajectory after the decision \(u\) has been realized. As was presented in Section 4, the fractional order \(v\) of the discrete difference \(\Delta Q^{v}\) might be used to fluently scan the space of the process states. The next component is connected with the necessity for the trajectory to omit the states of set \(S_{N}\), i.e., it takes into account the consequences of the decision \(u\) from the due date's limitations point of view. Let \(L(x^{\prime},t^{\prime})\) estimate the minimal distance between the new state \((x^{\prime},t^{\prime})\) and the subset of not-admissible set \(S_{N}\) that is reachable from the state \((x^{\prime},t^{\prime})\). The estimation is defined in detail in [16; 26]. Thus, involving the GL fractional derivative, the local criterion is of the form:  
\[q(u,x,t)=\Delta Q^{v}(u,x,t)+\widehat{Q}(u,x,t)+a\frac{1}{L(u,x^{\prime},t^{ \prime})}, \tag{17}\]  
where \(a\) is a parameter that is being changed during consecutive experiments (trajectory generations). The value of criterion \(q\) is computed for each \(u\in U_{p}(x,t)\). This decision \(u^{*}\) for which the criterion value is minimal is chosen. Then, the new state \((x,t^{\prime})=f(u^{*},x,t)\) is generated and the new best decision is chosen. If a newly generated trajectory is admissible and, for most of its states, the distances to the set \(S_{N}\) are relatively big, the parameter \(a\) can be decreased. In such a situation, the role of optimization compounds is enlarged. On the contrary, when the generated trajectory is not admissible, the parameter \(a\) should be increased because then the greater emphasis should be put on the due date's limitations. In a typical process of optimizing logistic tasks for unmanned aircraft, the process is affected by problems related to downtime (e.g., vessel failures or delays in flight tasks), continuous control of fleet vessels, their availability, optimal task planning, optimal use of the available capacity, and available airspace capacity. In current air task planning systems (such as the European ATM Master Plan), this is still a manual process that is supported by additional technical means. Intelligent technology is used to control the process of executing dynamically incoming air tasks by a fleet of drones that move concurrently according to the distribution of nodal points--individual air destinations. For such a use of algebraic-logical description, the following assumptions are defined:  
* The task of controlling the process of executing dynamically incoming aerial tasks by a fleet of drones that move concurrently according to the distribution of nodal points--individual aerial destinations--is defined. * The aviation destinations for individual tasks can dynamically appear in the system, according to the properties of the process state vector. * The drone fleet consisted of six unmanned aerial vehicles (UAV1-UAV6). * The UAVs take off and land at a dedicated airport; the nodal points are known. * The tasks to be executed arrive in real time and are respectively defined as \(T1,T2,\ldots,Tn\), having the structure of single-stage and multi-stage target nodes, respectively. * The execution of the assigned individual tasks for the UAV is carried out, respectively, at specific time intervals depending on the currently prevailing conditions and external interference, as well as taking into account the conditions related to possible dangerous/collision situations of the UAV. A number of studies have been conducted for different trajectory classes, number of drones and tasks. The example presented below is for a fleet of four drones that had to perform a set of 21 random tasks for a single-stage trajectory (i.e., the number of target nodes is 1).