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Mathematics
Feasible regions and resource optimization
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Drag a production plan and adjust resources and benefits to explore linear inequalities and optimal solutions.
Map of production plans
Drag P to try a plan
MaterialTimeMinimumFeasible regionEqual benefit↗More benefit
F = 3x + 4y → max
Model and consumption rates
2x + y ≤ 18
x + 2y ≤ 18
x + y ≥ 0
x ≥ 0 · y ≥ 0
A workshop blends products A and B. Choose quantities that fit its resources and maximize benefit.
1 kg A: 2 kg material · 1 hour. 1 kg B: 1 kg material · 2 hours.
P (4; 4)Feasible
F = 28 units
Material12 / 18 kg
Remaining: 6 kg
Time12 / 18 hours
Remaining: 6 hours
Total output8 kg
Minimum: 0 kg. Order met
Feasible vertices
Select a vertex to try it. Compare benefits at the vertices of a nonempty region.
Boundary lines belong to the region because ≤ and ≥ include equality.
Feasible.
Model notes
- A fictional workshop makes divisible quantities of A and B in kg. Each kg of A uses 2 kg material and 1 hour; each kg of B uses 1 kg material and 2 hours.
- Resources and benefits are linear. Consumption rates include assumed waste. Fixed costs, changing prices and integer restrictions are excluded. Benefit uses arbitrary units, not a business forecast.
- The feasible region satisfies 2x + y ≤ M, x + 2y ≤ T, x + y ≥ D and x, y ≥ 0. Finite resources bound every nonempty feasible region.
- A linear objective over a nonempty bounded polygon has an optimal vertex. An entire edge can also be optimal; if both benefit coefficients are zero, every feasible plan is optimal.
- Trial plans use 0.1 kg steps; the solver does not round intersections. Displays round to two decimal places. Objective lines show equal benefit, not production paths.