Question

MINIMIZATION BY THE SIMPLEX METHOD

convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method.

1>.

Minimize z = 6x₁ + 8x₂

subject to 2x₁ + 3x₂ ≥ 7

4x₁ + 5x₂ ≥ 9

x₁, x₂ ≥ 0

2>.

Minimize z = 4x₁ + 3x₂

subject to x₁ + x₂ ≥ 10

3x₁ + 2x₂ ≥ 24

x₁, x₂ ≥ 0

Answer 1

subject to

Iteration-1		Cj	6	8	0	0	M	M
B	CB	XB	x1	x2	S1	S2	A1	A2	MinRatio XB/x2
A1	M	7	2	3	-1	0	1	0	7/3=2.3333
A2	M	9	4	(5)	0	-1	0	1	9/5=1.8→
Z=16M		Zj	6M	8M	-M	-M	M	M
		Zj-Cj	6M-6	8M-8↑	-M	-M	0	0

Positive maximum Zj-Cj is 8M-8 and its column index is 2

Minimum ratio is 1.8 and its row index is 2

The pivot element is 5.

Entering =x2, Departing =A2,

Iteration-2		Cj	6	8	0	0	M
B	CB	XB	x1	x2	S1	S2	A1	MinRatio XBS2
A1	M	8/5	-2/5	0	-1	(3/5)	1	(8/5)/(3/5)=8/3=2.6667→
x2	8	9/5	4/5	1	0	-1/5	0	---
Z=8M/5+72/5		Zj	-2M/5+32/5	8	-M	3M/5-8/5	M
		Zj-Cj	-2/M5+2/5	0	-M	3M/5-8/5↑	0

Positive maximum Zj-Cj is 3M/5-8/5 and its column index is 4

Minimum ratio is 2.6667 and its row index is 1.

The pivot element is 35.

Entering =S2, Departing =A1

Iteration-3		Cj	6	8	0	0
B	CB	XB	x1	x2	S1	S2	MinRatio
S2	0	8/3	-2/3	0	-5/3	1
x2	8	7/3	2/3	1	-1/3	0
Z=56/3		Zj	16/3	8	-8/3	0
		Zj-Cj	-2/3	0	-8/3	0

all

optimal solution is arrived

.

.

.

.

.

subject to

Iteration-1		Cj	4	3	0	0	M	M
B	CB	XB	x1	x2	S1	S2	A1	A2	MinRatio XB/x1
A1	M	10	1	1	-1	0	1	0	10/1=10
A2	M	24	(3)	2	0	-1	0	1	24/3=8→
Z=34M		Zj	4M	3M	-M	-M	M	M
		Zj-Cj	4M-4↑	3M-3	-M	-M	0	0

Positive maximum Zj-Cj is 4M-4 and its column index is 1

Minimum ratio is 8 and its row index is 2.

The pivot element is 3.

Entering =x1, Departing =A2

Iteration-2		Cj	4	3	0	0	M
B	CB	XB	x1	x2	S1	S2	A1	MinRatio XBx2
A1	M	2	0	(1/3)	-1	1/3	1	2/(1/3)=6→
x1	4	8	1	2/3	0	-1/3	0	8/(2/3)=12
Z=2M+32		Zj	4	M/3+8/3	-M	M/3-4/3	M
		Zj-Cj	0	M/3-1/3↑	-M	M/3-4/3	0

Positive maximum Zj-Cj is M3-13 and its column index is 2

Minimum ratio is 6 and its row index is 1.

The pivot element is 13.

Entering =x2, Departing =A1,

Iteration-3		Cj	4	3	0	0
B	CB	XB	x1	x2	S1	S2	MinRatio
x2	3	6	0	1	-3	1
x1	4	4	1	0	2	-1
Z=34		Zj	4	3	-1	-1
		Zj-Cj	0	0	-1	-1

all

optimal solution is arrived