# Generalization of ALMM Based Learning Method for Planning and Scheduling
## 1 Introduction  
Curiosity is the main motivation for any human, a researcher in particular. Whenever a new method or algorithm is developed, multiple questions emerge. Could the method be applied under weaker assumptions, thus enhancing the area for its use? Could it be improved? In which ways? How does the method relate to the ones that came earlier? This paper refers to a learning method (a metaheuristic) called the ALMM Based Learning method. The objective of the method is to solve discrete (combinatorial) optimization problems, especially NP-hard ones. This paper presents further results of research involving the method and thus provides answers to the questions above. The method utilizes a special multistage decision process modeling paradigm referred to as the Algebraic Logical Metamodel of Multistage Decision Processes (abbreviated as ALMM of MDP and finally ALMM), which is presented in Section II. Based on the ALMM paradigm, one can develop mathematical models for discrete optimization problems--the so-called AL models. Even though a problem model may be known, its analytic solution is not available. Moreover, as the model has a recursive character, it is difficult to infer decision consequences by more than a single step. Initial work related to attempts to use algebraic logic models to optimize discrete process control have been proposed in [1, 2]. They used the concept of a simplified algebraic-logical description of the properties of the process that could be used to control its course. This approach had a strong limitation because it did not define a mathematical model of the process, but only relationships of its properties that could be used to make decisions controlling it [3, 4, 5]. In this paper, the authors define a generalized mathematical notation of the process model described in the ALMM technology, which takes into account the new mechanism of the learning strategy.  
The objective of this paper is twofold:  
* To present new generalizations of ALMM Based Learning methods involving a fractional derivative mechanism;
* To present essential remarks related to a comparison of Reinforcement Learning vs the ALMM Based Learning method.  
Many types of learning have been explored for some discrete optimization problems [6; 7; 8; 9; 10; 11; 12; 13; 14], especially scheduling: note learning, inductive learning, neural network learning, case-based learning, classifier systems, and others. Particular methodologies offer positive and negative features. However, none of the mentioned learning concepts use a mathematical model of a problem to be solved. The novelty of the machine learning presented in this paper is the fact that this method is based on a special mathematical model. The paper is constructed as follows. The Algebraic Logical Metamodel of Multi-stage Decision Process is presented in Section 2. Section 3 presents definitions of some criterion properties that are used in further discussion. An example that is an extension of the proposed learning method strategy is recalled in Section 4. Generalizations for the prime ALMM Based Learning method, as well as essential comments on a comparison of Reinforcement Learning with ALMM Based Learning, are given, respectively, in Sections 5 and 7. In Section 6, an example is presented.