CASE STUDY $2$: OPTIMIZING HEALTHCARE DELIVERY WITH DISCRETE STRUCTURES Scenario: You have been hired by a healthcare company to optimize the delivery of medical supplies and improve scheduling for mobile health units. The company operates several mobile health units that visit various clinics and rural locations, providing healthcare services. They need to ensure optimal routing for the units, efficient scheduling for patient appointments, and reliable supply chain management for medical supplies. Data Sources: Kaggle: Healthcare data, patient appointments, and medical supply chain data. OpenStreetMap: Geographical data for mobile unit routes between health clinics. Government Data Portals: Public healthcare statistics and travel times. Case Study $2$ Tasks Task $1$: Mobile Unit Scheduling Using Set Theory The company wants to optimize the mobile health unit schedules to ensure they visit the maximum number of clinics in a day while considering different clinic opening hours and vehicle availability. Data Source: Scheduling and clinic availability data from Kaggle or government health portals. Questions: $1.$ Set Operations: a$)$ Define sets for clinic opening hours and vehicle availability. Use set operations $($union$,$ intersection, etc.$)$ to determine optimal schedules for the mobile units. Note: Provide proper justification for the selected set theory operation$($s$).2.$ Relations: a$)$ Establish a relation between the mobile units and the clinics and analyze the constraints. Task $2$: Counting Principles in Patient Appointment Scheduling The company wants to optimize appointment schedules to minimize patient wait times and reduce congestion at clinics. Data Source: Appointment data and patient visit records from Kaggle. Questions: $1.$ Combinations and Permutations: a$)$ Determine how many different ways patient appointments can be scheduled based on clinic capacity and time slots. $2.$ Inclusion$-$Exclusion Principle: a$)$ Apply the inclusion$-$exclusion principle to avoid overlapping appointments at clinics where multiple mobile units operate. Task $3$: Routing Medical Supplies Using Graph Theory The healthcare company needs to optimize delivery routes for medical supplies to different clinics. Each clinic has varying demand levels and distance from the central warehouse. Data Source: Delivery route data from OpenStreetMap, combined with demand data from Kaggle. Questions: $1.$ Minimum Spanning Tree: a$)$ Use the minimum spanning tree to optimize the routes for medical supply deliveries, ensuring minimal travel time while covering all clinics. Note: Provide proper justification for the selected algorithm to find the minimum spanning tree. $2.$ Shortest Path Algorithm: a$)$ Apply Dijkstra$$s algorithm to find the fastest route for high priority supplies. Note: Algorithm$($s$)$ other than Dijkstra$$s may also be used with proper justification provided. Task $4$: Probability of Service Interruptions Traffic delays and unpredictable events such as vehicle breakdowns can disrupt the mobile units$$ schedules. The company wants to assess the probability of service interruptions. Data Source: Vehicle incident data and traffic statistics from Kaggle or government data portals. Questions: $1.$ Basic Probability: a$)$ Calculate the probability that a mobile unit will be delayed due to traffic or vehicle breakdowns based on historical data. $2.$ Conditional Probability: a$)$ Use Bayes$$ Theorem to determine the likelihood that a mobile unit experienced a breakdown if it arrives significantly late to a clinic. Learning Outcome: Students will apply set theory, counting principles, graph theory, and probability to optimize healthcare delivery, using real$-$world data from Kaggle, OpenStreetMap, and government data portals.