Question

Solve the following LPP:

Minimize z = 4x + 2y

Subject to 3x + y ≥ 27, x + y ≥ 21, x + 2y ≥ 30, x ≥ 0, y ≥ 0

Answer 1

We first draw the lines AB, CD and EF whose equations are 3x + y = 27, x + y = 21, x + 2y = 30 respectively.

Line	Equation	Points on the X-axis	Points on the Y-axis	Sign	Region
AB	3x + y = 27	A(9, 0)	B(0, 27)	≥	non-origin side of line AB
CD	x + y = 21	C(21, 0)	D(0, 21)	≥	non-origin side of line CD
EF	x + 2y = 30	E(30, 0)	F(0, 15)	≥	non-origin side of line EF

The feasible region is XEPQBY which is shaded in the graph.

The vertices of the feasible region are E(30, 0), P, Q and B(0, 27).

P is the point of intersection of the lines

x + 2y = 30       ....(1)

and x + y = 21   ....(2)

On subtracting, we get

y = 9

Substituting y = 9 in (2), we get

x + 9 = 21

∴ x = 12

∴ P is (12, 9)

Q is the point of intersection of the lines

x + y = 21         ....(2)

and 3x + y = 27    ....(3)

On subtracting, we get

2x = 6       

∴ x = 3

Substituting x = 3 in (2), we get

3 + y = 21   

∴ y = 18

∴ Q is (3, 18)

The values of the objective function z = 4x + 2y at these vertices are

z(E) = 4(30) + 2(0) = 120 + 0 = 120

z(P) = 4(12) + 2(9) = 48 + 18 = 66

z(Q) = 4(3) + 2(18) = 12 + 36 = 48

z(B) = 4(0) + 2(27) = 0 + 54 = 54

∴ z has minimum value 48, when x = 3 and y = 18.