# Generalization of ALMM Based Learning Method for Planning and Scheduling
## 2 Algebraic-Logical Meta-Model of Multistage Decision Processes  
The discrete process, which is implemented in the form of a sequence of control decisions, is an effective and practical approach that allows for monitoring various production processes. The key problem in constructing models of such process control is the need to have expert knowledge regarding the nature of such a process--whether it is deterministic or stochastic--but also the degree of their adjustment to real systems and the related confidence level of the conducted steering of this system. In this paper, the authors propose a mechanism of abstract mathematical modeling to describe a deterministic process using the approach paradigm named ALMM of MDP (abbreviated as ALMM). The idea of the ALMM paradigm was proposed and developed by [PERSON] in [1] and recalled in many other papers [5; 15; 16]. It has also been put to use in multiple cases [1; 17; 18; 19; 20; 21; 22]. Based on ALMM, the formal models, the so-called AL models, may be established for a very broad class of discrete optimization problems from a variety of application areas (especially for the modeling and control of discrete manufacturing and logistics processes), thus yielding to the meta-model designation. The description and definition of the ALMM paradigm, cited in the aforementioned articles, are cited below. \"ALMM is a general model development paradigm for deterministic problems, for which solutions can be presented as a sequence (or a set) of decisions, usually complex ones (i.e., composed of some single decisions). It facilitates convenient representation of all kinds of information regarding the problem to be solved, in particular the information defining a structure of states and decisions, algorithm used to generate consecutive states and various temporal relationships and restrictions of the problem. Furthermore, ALMM enables us to define various problem properties, in particular ones for which a particular heuristic method may be applied\". ALMM provides a structured way of recording knowledge of the goal and all relevant restrictions that exist within the problems modeled. Using this paradigm, the author has provided, i.e., in [3; 16], the definitions of two base types of multistage decision processes: a common process (cMDP) and a dynamic process (MDDP). The definition of MDDP quoted below refers to processes wherein both the constraints and the transition function depend on time. Therefore, the concept of the so-called \"generalized state\" has been introduced, defined as a pair containing both the state and the time instant. Definition 1: The multistage dynamic decision process is a process that is specified by the sextuple \(MDDP=(U,S,s_{0},f,S_{N},S_{G})\), where \(U\) is a set of decisions, \(S=X\times T\) is a set of generalized states, \(X\) is a set of proper states, \(T\subset\mathfrak{R}+U\{0\}\) is a subset of non-negative real numbers representing the time instants, \(f:U\times S\to S\) is a partial function called a transition function,(it does not have to be determined for all elements of the set \(U\times S\)), \(s_{0}=(x_{0},t_{0})\), \(S_{N}\subset S\) and \(S_{G}\subset S\) are, respectively, an initial generalized state, a set of not admissible generalized states, and a set of goal generalized states, i.e., the states in which we want the process to take place at the end. Subsets \(S_{G}\) and \(S_{N}\) are disjoint, i.e., \(S_{G}\cap S_{N}=\varnothing\)._  
_The transition function is defined by means of two functions, \(f=(f_{x},f_{t})\), where \(f_{x}\), \(f_{t}\) determine the next state and the next time instant, respectively. It is assumed that the difference \(\Delta t=f_{t}(u,x,t)-t\) has a value that is both finite and positive. Because not all decisions defined formally make sense in certain situations, the transition function \(f\) is defined as a partial one.