Question

Find the optimum solution to the following LP using the Simplex Algorithm. Use Big M method.

??? ?=5?1+2?2+?3 ??
?1 + ?2 ≤ 6
?1 − ?2 + 2?3 ≤ 2

3?1 + 2?2 = 12

?1,?2,?3 ≥0

Answer 1

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Solution:

Problem is

Max Z = 5 x1 + 2 x2 + x3

subject to

x1 + x2 ≤ 6 x1 - x2 + 2 x3 ≤ 2 3 x1 + 2 x2 = 12

and x1,x2,x3≥0;

The problem is converted to canonical form by adding slack, surplus and artificial variables as appropiate

1. As the constraint-1 is of type '≤' we should add slack variable S1

2. As the constraint-2 is of type '≤' we should add slack variable S2

3. As the constraint-3 is of type '=' we should add artificial variable A1

After introducing slack,artificial variables

Max Z = 5 x1 + 2 x2 + x3 + 0 S1 + 0 S2 - M A1

subject to

x1 + x2 + S1 = 6 x1 - x2 + 2 x3 + S2 = 2 3 x1 + 2 x2 + A1 = 12

and x1,x2,x3,S1,S2,A1≥0

Iteration-1		Cj	5	2	1	0	0	-M
B	CB	XB	x1	x2	x3	S1	S2	A1	MinRatio XBx1
S1	0	6	1	1	0	1	0	0	61=6
S2	0	2	(1)	-1	2	0	1	0	21=2→
A1	-M	12	3	2	0	0	0	1	123=4
Z=-12M		Zj	-3M	-2M	0	0	0	-M
		Zj-Cj	-3M-5↑	-2M-2	-1	0	0	0

Negative minimum Zj-Cj is -3M-5 and its column index is 1. So, the entering variable is x1.

Minimum ratio is 2 and its row index is 2. So, the leaving basis variable is S2.

∴ The pivot element is 1.

Entering =x1, Departing =S2, Key Element =1

Iteration-2		Cj	5	2	1	0	0	-M
B	CB	XB	x1	x2	x3	S1	S2	A1	MinRatio XBx2
S1	0	4	0	2	-2	1	-1	0	42=2
x1	5	2	1	-1	2	0	1	0	---
A1	-M	6	0	(5)	-6	0	-3	1	65=1.2→
Z=-6M+10		Zj	5	-5M-5	6M+10	0	3M+5	-M
		Zj-Cj	0	-5M-7↑	6M+9	0	3M+5	0

Negative minimum Zj-Cj is -5M-7 and its column index is 2. So, the entering variable is x2.

Minimum ratio is 1.2 and its row index is 3. So, the leaving basis variable is A1.

∴ The pivot element is 5.

Entering =x2, Departing =A1, Key Element =5

Iteration-3		Cj	5	2	1	0	0
B	CB	XB	x1	x2	x3	S1	S2	MinRatio
S1	0	1.6	0	0	0.4	1	0.2
x1	5	3.2	1	0	0.8	0	0.4
x2	2	1.2	0	1	-1.2	0	-0.6
Z=18.4		Zj	5	2	1.6	0	0.8
		Zj-Cj	0	0	0.6	0	0.8

Since all Zj-Cj≥0

Hence, optimal solution is arrived with value of variables as :
x1=3.2,x2=1.2,x3=0
Max Z=18.4

PLEASE RATE THE ANSWER, I have spent 2 hours to solve and format !!

PLEASE RATE THE ANSWER, I have spent 2 hours to solve and format !!

PLEASE RATE THE ANSWER, I have spent 2 hours to solve and format !!