Optimize within a feasible polygon
Why can checking a few vertices solve this linear production problem?
Entering midway? An objective measures what to improve; constraints define allowed choices. Feasible means all constraints hold.
Start with a concrete question
Produce real-valued amounts x,y with capacity, product caps and credits per unit. This is a continuous linear program.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
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What do you expect?
Predict before calculating: Why can checking a few vertices solve this linear production problem?
Change one thing. Inspect what happens.
- List the constraints and sketch their common region.
- Evaluate every feasible vertex at initial settings.
- Change objective price while holding the region fixed.
- Find a tie and explain the optimal connecting segment.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Feasible polygon and selected optimum
Horizontal: x production units. Vertical: y production units.
- 1. feasible boundary (solid line)
- 2. optimal vertex (points)
Axes use the displayed ranges.
Read plotted values as a table
| Series | x production units | y production units |
|---|---|---|
| feasible boundary | 0 | 0 |
| feasible boundary | 4 | 0 |
| feasible boundary | 4 | 2 |
| feasible boundary | 3 | 3 |
| feasible boundary | 0 | 3 |
| feasible boundary | 0 | 0 |
| optimal vertex | 4 | 2 |
Follow the calculation
- Constraints: x≥0, y≥0, x≤4, y≤3, x+y≤capacity.
- Objective: maximize 3x+2y.
- Feasible vertex (0,0) → 0 credits.
- Feasible vertex (4,0) → 12 credits.
- Feasible vertex (4,2) → 16 credits.
- Feasible vertex (3,3) → 15 credits.
- Feasible vertex (0,3) → 6 credits.
- The best listed vertex is unique under this objective.
For this bounded polygon, a linear objective attains a maximum at a vertex. That justifies checking vertices; it is not a rule for arbitrary nonlinear objectives.
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Explain the mechanism
Optimize within a feasible polygon
Produce real-valued amounts x,y with capacity, product caps and credits per unit. This is a continuous linear program.
Concepts to recall
An objective measures what to improve; constraints define allowed choices. Feasible means all constraints hold.
Understand the mechanism
Initially x+y≤6, x≤4, y≤3, and objective 3x+2y gives optimum (4,2), value 16.
A linear objective on a nonempty bounded polygon reaches its maximum at a vertex. Along an edge it changes linearly or stays constant.
At equal prices the objective depends only on x+y, so multiple allocations along the capacity edge can tie. Integer production would require an integer model.
Symbols and calculation
Maximize p x+2y subject to x,y≥0, x≤4, y≤3, x+y≤C. p is credits per x unit; C is shared capacity.
Worked example
Constraints: x≥0, y≥0, x≤4, y≤3, x+y≤capacity.
Objective: maximize 3x+2y.
Feasible vertex (0,0) → 0 credits.
Feasible vertex (4,0) → 12 credits.
Feasible vertex (4,2) → 16 credits.
Feasible vertex (3,3) → 15 credits.
Feasible vertex (0,3) → 6 credits.
The best listed vertex is unique under this objective.
Your investigation
List the constraints and sketch their common region.
Evaluate every feasible vertex at initial settings.
Change objective price while holding the region fixed.
Find a tie and explain the optimal connecting segment.
Limits and interpretation
- These are bounded teaching models with explicit assumptions, not evidence of a real-world intervention.
Try a fresh case
For x+y≤5, 0≤x≤3, 0≤y≤4, maximize 4x+2y. What is the maximum?
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Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Explain a fresh small numerical case with a diagram or table and an argument.
College investigation
Implement the method, test a boundary or counterexample, and justify its assumptions.
Workplace investigation
Use a small decision scenario, state units and constraints, and test how the recommendation changes.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.