# Heuristic Method for Decision-Making in Common Scheduling Problems
## 4 Solution Graph Searching Based on ALMM  
The approach presented in this paper uses the state space, in particular the state graph. It belongs to a group of solutions that designate a solution based on a multi-stage decision-making process and consists of generating a finite sequence trajectory. In particular, it can be used for problems where the quality criterion is additively separable and monotonically growing along the trajectory. The task of searching for a solution based on an algebraic-logical meta-model methodology is to find a sequence of decisions that define the admissible trajectory. On the other hand, the task of the process control optimization is to find such a sequence of admissible decisions for which the quality is the best. The trajectory is equivalent to finding the path in the generalized state space that joins the initial state \(s_{0}=(x_{0},t_{0})\) with the state belonging to the finish set, that is, the set of goal states \(S_{G}\) or set of non-admissible states \(S_{N}\). The notion of semimetrics in the state space, proposed by [PERSON] [27; 28], is used to evaluate states and their belonging to characteristic states. Thus, the distances between the current state and the states of the characteristic sets (goal, non-admissible, or favorable) are determined during a single trajectory generation. Moreover, each trajectory, whether admissible or non-admissible, is analyzed and the knowledge gained is used to improve control while generating successive trajectories. Thus, knowledge of the control process is obtained from the analysis of the process description, each state of a single trajectory, and the results of a previously obtained solution (trajectory).  
It should be noted that the generalized state graph has a structure similar to the tree structure. This is because even if for two generalized states \(s_{1}=(x,t_{1})\) and \(s_{2}=(x,t_{2})\) the proper state \(x\) is the same, then the generalized state will be different if there is even a very small time difference between \(t_{1}\) and \(t_{2}\).  
Although the algebraic-logical model provides an examination of all solutions (because a set of possible decisions in each state can be determined), the approach described below facilitates constructing not the whole tree but only its most perspective part. The solution to the problem can be searched for in two ways. First, a successive trajectory can be generated independently, using information derived from the analysis of the obtained solutions to modify, for example, simulation parameters. The algorithm using that method of generating a single trajectory with the idea of learning was proposed in [28] and developed in [32; 33]. Secondly, a new trajectory can be created by correcting the final part of previously generated trajectories [34]. It is important then to determine the rule for selecting this process state from which the new final part of the trajectory will be generated. This state is selected on the basis of the information obtained previously, in the form of specially chosen and calculated parameters in each state. A general method using a second way of finding a solution is the subject of this paper.