Slide the profit line
"Slide the profit line" is a free, interactive Class 12 Maths simulation for JEE, built on the lesson "Multiple optimal solutions" in Linear Programming. Every plan that earns exactly k rupees lies on the line 150x + qy = k, where q is the profit on a cookie tray. Raise k and the line slides outwards, parallel to itself. The orange part is the plans that earn k and are also allowed. Level lines of Z are parallel. The optimum is where the last level line leaves the region: a corner, or a whole edge when the level lines run parallel to that edge.
From the lesson Multiple optimal solutions in Linear Programming. Change the values and watch what happens.
The idea behind it
NCERT §12.2.2, Example 3
- Sometimes two corners give the same best value. Then every point of the edge joining them gives it too, so the problem has infinitely many optimal solutions.
- This happens when the lines Z = k run parallel to that edge: the sliding line leaves the region along a whole side, not at a single point.
- Example: Z = 3x + 9y where x ≤ y, x + y ≥ 10 and x + 3y ≤ 60, with x, y ≥ 0. Z is 90 at corner A(0, 10), 60 at B(5, 5), and 180 at both C(15, 15) and D(0, 20).
- So the minimum is 60 at B(5, 5), and the maximum 180 is reached at both C and D, and at every point of CD, such as (7.5, 17.5). Here 3x + 9y is 3 times x + 3y, which is exactly the boundary line through C and D.
- The same holds for minima: two corners with the same smallest value make the whole edge between them optimal.
Questions
2 answers
- What does the "Slide the profit line" simulation show?
- Every plan that earns exactly k rupees lies on the line 150x + qy = k, where q is the profit on a cookie tray. Raise k and the line slides outwards, parallel to itself. The orange part is the plans that earn k and are also allowed.
- Is this simulation free?
- Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Multiple optimal solutions" in Linear Programming (Class 12 Maths).
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