On connected sub-tree with fixed nodes in Cayley tree of Order 2

In this paper we found an exact formula for a finite sub-tree counting problem. Note that the formulas, which correspond to two extremal cases, are Catalan Triangle introduced by Shapiro and ballot Catalan triangles. The general formula could be expressed as a linear combination of these Catalan triangles

Using Dimensional Aware Genetic Programming to find interpretable Dispatching Rules for the Job Shop Scheduling Problem

Dispatching Rules (DRs) have been used in several applications in manufacturing systems. They assign priority to jobs in a queue choosing the next job to be executed. As they are challenging to design, genetic programming (GP) is being used to find better performative DRs. In GP, several different DRs are evolved, and due to some operations and selection processes inspired in nature, the DRs improve. However, little research has been done in trying to reach small and interpretable DRs. Usually, these generated expressions tend to become extremely large, with a couple of hundred terms or more. This work will innovate by using CFG (context-free grammars) methods, particularly CFG-GP and GE (Grammar Evolution), for reaching DRs which are dimensional aware. These methods will be compared as they have several distinct characteristics and were never used for this problem. The objective is that by forcing the syntax of the DRs to be correct, it will be possible to reach smaller and more interpretable DRs. Furthermore, an enumerator was made that found the best possible expression for a small DRs size, which will serve as a baseline to evaluate how well the different algorithms can explore these spaces and give the best possible DRs for a specific size. The results show a significant performance improvement in using DAGP methods for this problem. Moreover, GP/GE and CFG-GP can explore the small DRs optimally or close to optimally, managing to find the best small DRs