# Generalization of ALMM Based Learning Method for Planning and Scheduling
## 4 Machine Learning Based on ALMM  
In the work [3], the authors proposed a machine learning method that used the changes dynamics of the \(\Delta Q\) criterion and the weighted sum of partial criteria to determine the local quality criterion \(q\). The learning process was reduced to determining a set of trajectories of process states leading to \(S_{G}\) states and sequentially searching for such control sequences \(u\), taking into account the selection of the weight vector \(a\). To avoid unnecessary redundancy, we recall briefly the basic form of the local criterion \(q(u,s)\) which became the following form:  
\[q(u,s)=a_{0}\big{(}\Delta Q(u,s)+\hat{Q}(u,s)\big{)}+a_{1}\varphi_{1}(u,s)+ \cdots+a_{i}\varphi_{i}(u,s)+\cdots+a_{n}\varphi_{n}(u,s), \tag{6}\]  
where \(\Delta Q(u,s)\) is the change of the criterion value as a result of decision \(u\), undertaken in the state \(s\), \(\hat{Q}(u,s)\) is the estimation of the quality index value for the final trajectory section after the decision \(u\) has been realized, \(\psi(u,s)\) is the components reflecting additional limitations or additional requirements in the space of states, \(i=1,2,\ldots,n\), \(a_{i}\) is the coefficient which defines the weight of \(i\)-th component \(\varphi_{i}(u,s)\) in the criterion \(q(u,s)\). The sum of weight coefficients \(a_{i}\) for \(i=1,2,\ldots,n\) is equal to \(1\). A role of the criterion components connected with considered subsets should be strengthened for the next trajectory, i.e., the weights (priorities) of these components increase. When the generated trajectory is admissible, the role of the components responsible for the trajectory quality can be strengthened, i.e., their weights can be increased. Based on the gained information, the local optimization task is being improved during simulation experiments. This process is treated as a learning or intelligent searching algorithm. As the \(q\) criterion coefficients change in consecutive iterations, the criterion assumes the following form:  
\[q^{(k)}(u,s)=a_{0}^{(k)}\Big{(}\Delta Q(u,s)+\widehat{Q}(u,s)\Big{)}+a_{1}^{( k)}\varphi_{1}(u,s)+\cdots+a_{i}^{(k)}\varphi_{i}(u,s)+\cdots+a_{n}^{(k)} \varphi_{n}(u,s), \tag{7}\]  
where \(k\) denotes the number of iteration. Typically, the same initial coefficient values are assumed. Changes to the \(a_{2},a_{3},\ldots,a_{n}\) coefficients depend on real maximum distances of the last generated trajectory from the reachable parts of the not-admissible state set \(S_{N}\), and the remaining special sets. In the work [24], the fractional back propagation algorithm was proposed (FBP), which uses the fractional order derivative according to Grunwald-Letnikov fractional derivative (GL) theory. Assuming the general form definition of the integer derivative and the fractional derivative, the GL derivative can be written:  
\[u_{0,s_{0}}\Delta_{u,s}^{v}Q(u,s)=\lim_{h\to 0}\frac{1}{h^{v}}\sum_{n=0}^{ \lfloor(u-u_{0}s-s_{0}/h)\rfloor}(-1)^{n}\binom{v}{n}Q(u_{i}-u_{i-nh},s_{i}-s_ {i-nh}), \tag{8}\]  
where \(\binom{v}{n}\) denotes the Newton binomial, \(v\) the order of the fractional derivative of basis function \(\varphi(u,s)\), \(u_{0}\), \(s_{0}\) the interval range, and \(h\) denotes the number of steps in the state space of the given process. Then, we can assume the backwards difference of the fractional order as \(u_{0,s_{0}}\Delta_{u,s}^{(v)}\), where \(v\in\mathbb{R}^{+}\):  
\[u_{0,s_{0}}\Delta_{u,s}^{(v)}=\sum_{n=0}^{\lfloor(u_{i}-u_{0}s_{i}-s_{0})/h \rfloor}b_{n}^{(v)}Q(u_{i}-u_{i-nh},s_{i}-s_{i-nh}), \tag{9}\]  
where the consecutive coefficients \(b_{n}^{(v)}\) are defined as follows:  
\[b_{n}^{(v)}= \begin{cases}1&\text{for $n=0$}\\ (-1)^{n}\frac{v(v-1)(v-2)...(v-n+1)}{n!}&\text{for $n=1,2,3,...,N$.}\end{cases}  \tag{10}\]  
\(N\) denotes the number of discrete measurements of \(Q(u,s)\). The \(\Delta Q(u,s)\) in Equation (7) is therefore true for the special case when \(v=1\) and \(N=1\).