Simplex method in prime how to use the constrains maximise s.t. constraints.

[Albert Chan]

(11-01-2023 02:42 AM)Anthony The Koala Wrote:  Minimise: C = 6x + 5y
Subject to:
x >= 0, y >= 0
2x + 4y >= 22
3x + 2y >= 20
4x + 5y >= 40

L0: C = 6x + 5y  --> slope = -1.2
L1: 2x + 4y = 22 --> slope = -0.5
L2: 3x + 2y = 20 --> slope = -1.5
L3: 4x + 5y = 40 --> slope = -0.8

Constraints slope are different than L0 slope, solution must be one of the vertices.
Graphically, we start with 0 = 6x + 5y, and "raise" it (increase C) until it hit a vertex.

Sort the slopes, smallest to biggest, we have {L2, L0, L3, L1}
Based on slope alone, vertex is intersection of L2 and L3

2*L2 - L3 --> 2x - y = 0 --> y = 2x
L2: 3x + 2y = 7x = 20    --> x = 20/7
L1: 2x + 4y = 10x = 200/7 ≈ 28.57 ≥ 22      ✔ (20/7, 40/7) is a true vertex

min(C) = 6x + 5y = 16x = 320/7 ≈ 45.71