Solving scheduling problems in practical environments, such as manufacturing facilities, can be a big challenge. Theoretical mathematical methods that aim to address these problems are oftentimes underutilized, both due to difficulties adapting them to the specific problem at hand, and limited mathematical expertise in an organization. The aim of this report is to address both of these issues by providing a step-by-step guide on the application of mathematical scheduling theory, including tools on dealing with difficult constraints. To showcase the effectiveness of this guide and its process, we provide a case study where the guide was applied to a manufacturer of generic medicine. Here we encountered many problems, such as non-standard constraints and large-sized data, but success- fully addressed each of them with a mathematical model capable of generating strong, practical solutions. While further improvements are definitely possible and encouraged, this research provides a strong proof of concept on how the gap between practice and theory can be addressed.

The Game of Cycles, invented by Francis Su (2020, p.51) is an impartial game played on a graph, where players take turns marking an edge according to a set of rules. Together with the game, there also came a conjecture that gives a condition for whether a specific position is winning or losing. Proving or disproving this conjecture is the main focus of this research, which we end up succeeding in by giving a counter-example, thus disproving the conjecture. We do this by first showcasing some relevant background knowledge from game theory in chapter 1. In chapter 2 we then introduce the Game of Cycles and its rules, as well as some of the previous results others have found. We continue in chapter 3 by creating a python script to brute-force the game for us and it is here that we find a counter-example to the main conjecture, of which we prove that it is indeed a counter-example. We close off with chapter 4 by looking at a simplification of the game, where it is played on trees instead of any graph. Here we prove that the main conjecture does hold for a special family of trees and state a conjecture for the solution of any tree.