# Generalization of ALMM Based Learning Method for Planning and Scheduling
## 3 Properties of Problems  
The vast majority of methods and algorithms described in the literature use (with or without overt declaration) some properties of criteria that facilitate the solving process. Some of these, defined based on the ALMM paradigm, have been presented in [3; 19] and then recalled in [23], are shortly recalled below. A broad class of criteria can be defined by recurrence and computed in parallel to the calculation of trajectories, the author named the said class as separable criterions (Definition 2). It is worth remembering, though, that these are not the only criterion classes that may be used. Let us denote \(P\)--a fixed multistage decision process, \(S^{P}\)--a set of all states of trajectories of the process, \(d(\tilde{s})\)--the number of the last state of a finite trajectory \(\tilde{s}\), \(\tilde{U}\)--a set of all decision sequences of the process \(P\), \(\mathfrak{R}\)--a set of real numbers.  
**Definition 2**.: _Criterion \(Q\) is separable for the process \(P\) if, for every decision sequence \(\tilde{u}\in,\tilde{U}_{t}\) can be recursively calculated as follows:_  
\[Q_{0}=const,\text{ in particular }Q_{0}=0 \tag{1}\] \[Q_{i+1}=f_{Q}(Q_{i},u_{i},s_{i})\text{ \ \ for \ \ }i=0,\text{ \ }1,\text{ \ }...,\text{ }d(\tilde{s})-1,\]  
_where \(Q_{i}\) for \(i>0\) denotes a partial value of criterion \(Q\) calculated for the \(i\)-th state of the considered trajectory, defined as follows:_  
\[Q_{i}=Q(\tilde{u}^{\prime}), \tag{2}\]  
_where \(\tilde{u}^{\prime}=(u_{0},u_{2},...,u_{i-1})\) is the initial part of the sequence \(\tilde{u}\), \(f_{Q}\) is some partial function \(f_{Q}:\mathfrak{R}\times\mathcal{U}\times S\rightarrow\mathfrak{R}\) such that:_  
\[Domf_{Q}=\Big{\{}(a,u,s)\in\mathfrak{R}\times\mathcal{U}\times S:s\in S^{P},u \in\mathcal{U}_{P}(s),a\in\mathfrak{R}\Big{\}}. \tag{3}\]  
_Separability is a property of an algorithm which calculates a quality criterion for a sequence of decisions \(\tilde{u}\), and thus for designated trajectory \(\tilde{s}\). The criterion is separable if we can calculate its value for the next state of a trajectory knowing only its value in the previous state and the decision taken at that time._  
_Particularly useful is the property of additive separability of criterion. Let \(Q\) be a separable criterion and a function \(\Delta Q\) denotes a change of criterion for two consecutive states on any trajectory, i.e., \(\Delta Q_{i}=Q_{i+1}-Q_{i}\)._  
**Definition 3**.: _Separable criterion \(Q\) is additive for a process \(P\), iff for each trajectory \(\tilde{s}\) of the process \(P\) and for each \(i=0,1,\ldots,d(\tilde{s})-1\) occurs:_  
\[f_{Q}(Q_{i},u_{i},s_{i})=Q_{i}+\Delta Q(u_{i},s_{i}), \tag{4}\]  
_i.e., \(\Delta Q\) depends on the state and decision only._  
**Definition 4**.: _Separable criterion \(Q\) changes multiplicatively for a process \(P\), iff for each trajectory \(\tilde{s}\) of the process \(P\) and for each \(i=0,1,\ldots,d(\tilde{s})-1\) the following is true:_  
\[f_{Q}(Q_{i},u_{i},s_{i})=Q_{i}\cdot\
u(u_{i},s_{i}), \tag{5}\]  
_where \(\
u(u,s)\) is a certain function depending on the decision and the state only [16]._