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

[Albert Chan]

(11-01-2023 07:32 PM)HPCarnace Wrote:  ... as dg1969 suggests it is better with "AND" (the feasible region is shown more clearly)

Still, with overlapping regions, it is very hard to spot the solution.
Even *after* we have the solution, confirming with plots may be tough.
This is especially true if slopes are close, which made polygon look like circle.

It may be better to plot something else ...

C = 6x + 5y --> y = (C-6x)/5
Elminate y from constraints, and solve for C

L1: 2x + 4y ≥ 22      → C ≥ 3.5x + 27.5
L2: 3x + 2y ≥ 20      → C ≥ -1.5x + 50
L3: 4x + 5y ≥ 40      → C ≥ 2x + 40

For vertices, inequality changed to equal sign. With x≥0, we have:
L1: C ≥ 27.5
L2: C ≤ 50
L3: C ≥ 40               → 40 ≤ C ≤ 50

If we plot below (V1,V2,V3), min(C) is the "lowest" intersection.
If a ball fall from the sky, min(C) is where it landed.

gnuplot> V1(x) = 3.5*x + 27.5
gnuplot> V2(x) = -1.5*x + 50
gnuplot> V3(x) = 2*x + 40
gnuplot> set terminal dumb
gnuplot> plot [0:5] [40:50] V1(x), V2(x), V3(x)

Code:

  50 ###-----------+-------------+-------------+-------------+------------$$
     +  #####      +             +             +             +V1(x) ****** +
     |       #####                                            V2(x)$###### |
     |            #####                                       V3(x) $$$$$$ |
  48 ++                #####                               $$$$           ++
     |                      #####                      $$$$                |
     |                           #####              $$$                    |
     |                                #####    $$$$$                       |
  46 ++                                   ######                          ++
     |                                  $$$$   #####                       |
     |                               $$$$           #####                 **
  44 ++                          $$$$                    #####         ***++
     |                       $$$$$                            #####  ***   |
     |                    $$$                                      ***##   |
     |                $$$$                                       ***   #####
  42 ++           $$$$                                         **         ++
     |         $$$$                                          **            |
     |     $$$$                                            **              |
     + $$$$        +             +             +         **  +             +
  40 $$------------+-------------+-------------+-------**----+------------++
     0             1             2             3             4             5

Even with crude ASCII plot, we get min(C) ≈ 46, from V2, V3 intersection.