Question

(1) Convert the following LPs to standard form: 22 (a) max z 3x1 + 2x2 s.t. 21 < 40 X1 + x2 < 80 2x1 + x2 < 100 X1, X2 > 0 (b) max z = 2x1 s.t. X1 – X2 <1 2x1 + x2 > 6 X1, X2 > 0 (c) max z = 3x1 + x2 s.t. 1 > 3 X1 + x2 < 4 2x1 – X2 = 3 X1, X2 > 0

Answer 1

The standard form required all constraints to be in '=' form. For '<=" constraints, this is obtained by adding a slack variable (s) to the LHS. For '>=' constraints, a surplus variable is subtracted instead.

Also, the RHS values must be >= 0. So, if there are any negative RHS, then both LHS and RHS will be multiplied by -1.

Also, all the decision variables will be >= 0. So, if there is any variable <= 0 declared, convert this to another non-negative variable by multiplying -1. If there is any free variable, replace it by the difference between two non-negative variables.

(a)

max z = 3x₁ + 2x₂ + 0s₁ + 0s₂ + 0s₃
s.t.
x₁ + s₁ = 40
x₁ + x₂ + s₂ = 80
2x₁ + x₂ + s₃ = 100
x₁, x₂, s₁, s₂, s₃ >= 0

(b)

max z = 2x₁ - x₂ + 0s₁ + 0s₂
s.t.
x₁ - x₂ + s₁ = 1
2x₁ + x₂ - s₂ = 6
x₁, x₂, s₁, s₂ >= 0

(c)

max z = 3x₁ + x₂ + 0s₁ + 0s₂
s.t.
x₁ - s₁ = 3
x₁ + x₂ + s₂ = 4
2x₁ - x₂ = 3
x₁, x₂, s₁, s₂ >= 0