Question

Write the following linear program in standard form. If your answer is zero, enter “0”. If the constant is "1" it must be entered in the box.

Max	12A	+	6B
s.t.
	A	−	2B	≤	420
	6A	+	8B	≤	1,300
	10A	−	2B	≤	250
A, B ≥ 0

	A	+	B	+	S1	+	S2	+	S3
s.t.
	A	−	B	+	S1
	A	+	B			+	S2
	A	−	B					+	S3
									A, B, S1, S2, S3

Answer 1

The standard form is obtained as follows :

• all variables involved are restricted to be non-negative.

• all constraints are equalities , with constant, non-negative right-hand sides.

Converting may require new variables and rearranging constraints:

• an inequality can be multiplied by −1 to get non-negative RHS.

• inequalities can be converted to equalities by adding or subtracting non-negative slack variables.

• Unrestricted variables can be dealt with by writing the variable as the difference of two new non-negative variables.

We have given the following problem :

Max Z = 12A + 6B

s.t.

A - 2B ≤ 420

6A + 8B ≤ 1300

10A - 2B ≤ 250

A, B ≥ 0

To convert to standard form, we introduce three new slack variables, s₁ ≥ 0 , s₂ ≥ 0 and s₃ ≥ 0.

The standard form of above problem is as follows :

Max Z = 12A + 6B + 0s_{1 +} 0s_{2 +} 0s₃

s.t.

A - 2B + s₁ = 420

6A + 8B + s₂ = 1300

10A - 2B + s₃ = 250

A, B, s₁, s₂ and s₃ ≥ 0.