consider a LP problem.. Please help?

luck

Consider a LP Problem

Max z = 4x₁ + 3x₂ + 2x₃

s.t.

2x₁ + x₂ + 3x₃ ≤ 10

x₁ + 3x₂ + 4x₃ ≤ 15

Xi ≥ 0

1. Solve the LP Problem

2. Determine the range of c₁ and c₃, for which the current basis remains optimal.

3. Determine the range of b₁ and b₂, for which the current basis remains optimal.

Please help me experts... gotta exam coming up in 2 days.

make it easier for me to understand please

Thank you

asimov

i take instead of ≤ , just =

and we have 2 even

2x₁ + x₂ + 3x₃ = 10

x₁ + 3x₂ + 4x₃ = 15

find the common line

multiple 1st equation with -3 and add to secod one

-6x₁ - 3x₂ - 9x₃ = -30

x₁ + 3x₂ + 4x₃ = 15

-------------------------------

-5x₁ - 5x₃ = -15

take x₁ =u , and divide by -5

u + x₃ = 3

solve for x₃

x₃ = 3 -u

plug both in equation 1

2x₁ + x₂ + 3x₃ = 10

2u + x₂ + 3(3-u) = 10

2u + x₂ + 9 -3u = 10

x₂ = 1 +u

now we have the common line

[X] = [x₁ , x₂ , x₃] = (0, 1 , 3) + u (1 , 1 , -1)

find the intercept points with the basis plane xy , xz and yz

for example for xy plane we set x₃=0

0 = 3 - u

u = 3

plug this in the line

[P] = (0,1,3) + 3( 1, 1, -1) = (3 , 4 , 0)

same way we find Q( 1,0,3) and R( -1,0,4)

R is not acceptable becouse Xi ≥ 0

now to find which one is max

plug P and Q in

z = 4x₁ + 3x₂ + 2x₃

for P we get 24 and for Q we get 9

the point P (3, 4, 0) is the answer