## Appendix D
_Choice of Data Structure_: Employs a consistent logging technique. _Complexity Reasoning_: Remains constant (O(1)) due to being a unique operation at the end of a process. Total (Excluding Constant Ops)

Overall Complexity Reasoning: The complexities stem from the repetitive process of cycling between segments and Points of Interest (POIs) (n\(\times\)m) together with the consistent operations within each cycle, leading to a mix of linear and logarithmic behaviors depending on the specific stage. Space complexity is the maximum memory needed to store all data needed for the implementation of this specific methodology. ## Appendix E

In this Appendix, a comprehensive analysis of mission planning and execution is presented. Additionally, a detailed and structured approach is provided for understanding, analyzing, and managing complex missions. This is achieved by breaking them down into their fundamental questions, themes, and components. Furthermore, a set of mathematical formulas is used to describe an explanatory implementation of the methodology for mission decomposition. _E1. Mission Analysis--The Rule of W(\({}^{\text{(j)}}\) Questions_

The Commanding Officers (COs) and the personnel at their Headquarters (HQ) examine the original mission in accordance with the Military Decision-Making Process (MDMP) before designating a mission statement to the participating actionable actors. This is a brief sentence or paragraph that might be properly explained by asking the following questions (Field Manual 101-5. Staff Organization and Operations. US Army, 1997):

* _Who_ will execute the mission (unit or organization),
* _What_ they will do during the mission (tactical mission essential task),
* _Where_ they will operate (area of operations, objective, grid coordinates),
* _When_ the operation will begin (by time or event) or what is the duration of the operation,
* _Why_ will the force conduct the operations (why is the mission necessary), and
* _What_ else could be added as information, known as the rule of: \[\text{W}^{\text{(j)}}\text{ questions, where j }\in\mathbb{N}^{\star}\] (1)

The content of the questions fully describes the initial missions including tasks, objectives, and purposes. It is obvious that the rule of questions can also be applied to any other participant actor, describing its sub-mission, which is associated with the initial mission. As previously indicated, missions consist of two fundamental components: tasks and purposes. Tasks pertain to \"who\", \"what\", \"where\", and \"when\", while purposes deal with \"why\" (Joint Publication (JP 5-0). Planning, Joint Operation. Chairman of the Joint Chiefs of Staff (CJCS), Department of US Army, 2017). Simple questions are \"who\", \"where\", and \"when\" in a goal statement. The questions \"what\" and \"why\" in a mission statement are both necessary. The question \"what\" is an impact task, typically quantifiable, that is stated using action verbs. A mission statement's \"why\" explains the objectives (or justifications) for the actionable actor to carry out the assignment. Because it describes the purpose of the work and puts it into context, it is crucial to mission commands and mission orders. If not articulated properly, the \"what\" and \"why\" questions might confuse subordinates and they are more important and difficult to write (Field Manual 3-90. (Book): Tactics. Headquarters Department of the Army U.S. CreateSpace Independent Publishing Platform, 2001 edition (15 April 2012)). _E2. Mathematical Approach of Thematic Decomposition of Mission_

The theme decomposition of the first mission, \"M0: W(j) questions\", should be clearly stated as it is evident that it is a tight structural analysis of division into sub-missions for each actionable actor. To fulfill the mission's objectives, duties, and goals, there are a number of prerequisites that must be understood.