📋 Activity Overview

Students model a Primary Health Centre allocating beds among maternity, fever, and emergency cases — setting up constraints as inequalities and finding the allocation that maximises patient outcomes.

💡 Teacher Tip

Ask: 'If your optimal solution gives 25 fever beds and 5 maternity, but a pregnant woman arrives during full capacity — what do you do?' The gap between optimised models and human values is one of the most important conversations in modern AI ethics.

🎯 Learning Objectives

  • ✓ Formulate a real-world resource allocation problem as a linear programme
  • ✓ Express constraints as linear inequalities
  • ✓ Solve graphically by identifying the feasible region and corner points
  • ✓ Interpret mathematical results in the context of healthcare decisions

🗂️ Materials Needed

PHC scenario cards Graph paper (large) Ruler and pencil Constraint equation worksheet Graphical LP template Calculator Coloured pencils

📌 Step-by-Step Instructions

Scenario (5 min) — 'Rampura PHC has 40 beds. Maternity cases need 3 days avg, fever cases 1 day avg, emergencies 5 days avg. You have 120 bed-days per week. Maternity contributes 8 health-points per bed, fever 3, emergency 12. Maximise total health-points. Emergency beds ≤ 10. At least 5 maternity beds at all times.'
Define Variables (5 min) — Let x = maternity beds, y = fever beds, z = emergency beds. x + y + z ≤ 40. 3x + y + 5z ≤ 120. z ≤ 10. x ≥ 5.
Simplify for Graphical Solution (5 min) — Fix emergency beds at maximum (z=10, 50 bed-days used). Now solve 2-variable LP: x + y ≤ 30, 3x + y ≤ 70, x ≥ 5.
Graph the Feasible Region (12 min) — Plot the boundary lines and shade the feasible region. Identify corner points.
Evaluate Objective at Corner Points (8 min) — Calculate health-points at each corner. Which allocation maximises health-points?
Real-World Interpretation (8 min) — 'The optimal mathematical solution says: 5 maternity, 25 fever, 10 emergency. Does this feel right for a real PHC? What constraints did we miss?'
Discuss Limitations (7 min) — Unmodelled realities: seasonal disease spikes, staff constraints, medicine availability. Why is the mathematical model a starting point, not a final answer?
Extension (10 min) — Add a new constraint: maternity beds must be at least 20% of total in monsoon season. How does the optimal point change?

🧠 CT Pillar Connections

Algorithmic Thinking
Linear programming is a formal algorithm: formulate, constrain, optimise, interpret. The graphical method is a precise procedure that always finds the answer at a corner point.
Abstraction
The PHC's complex reality (staff rosters, medicine supplies, patient dignity) is abstracted to 3 variables and 4 inequalities — powerful enough to give insight, simple enough to solve.
Decomposition
The problem decomposes into: define variables → write constraints → plot feasible region → evaluate objective. Each step is a tractable sub-problem.

💬 Discussion Questions

  • What real-world factors would you add to make this model more realistic?
  • How do hospitals use algorithms to schedule surgeries and allocate ICU beds?
  • Is it ethical to optimise healthcare purely for 'health-points'? What does that assumption hide?
  • Linear programming assumes smooth, continuous values. What breaks if we only work with whole beds?