Question

Question 1: Consider the following LP model.

max ? = 4?1 + 4?2

s.t. 3?1 − ?2 ≤ 9

?1 + ?2 ≤ 7

5?2 ≤ 25

?1 ≥ 0, ?2 ≥ 0

Part a) Find the optimal solution for the above model using simplex technique.

Part b) Find the shadow prices (optimal dual variables) for the model.

Part c) If you could buy an additional unit of the first resource for a cost of 5/2, would you do this? Why?

Answer 1

(a)

Standard form:

max ? = 4?1 + 4?2 + 0s1 + 0s2 + 0s3

s.t. 3?1 − ?2 + 1s1 = 9

?1 + ?2 + 1s2 = 7

5?2 + 1s3 = 25

?1, ?2, s1, s2 ≥ 0

Iteration-1		Cj	4	4	0	0	0
B	CB	XB	x1	x2	S1	S2	S3	Min Ratio XB / x1
S1	0	9	(3)	-1	1	0	0	9 / 3=3→
S2	0	7	1	1	0	1	0	7 / 1=7
S3	0	25	0	5	0	0	1	---
Z=0		Zj	0	0	0	0	0
		Zj-Cj	-4↑	-4	0	0	0

Negative minimum Zj-Cj is -4 and its column index is 1. So, the entering variable is x1.

The minimum ratio is 3 and its row index is 1. So, the leaving basic variable is S1.

So, the pivot element is 3.

Entering =x1, Departing =S1, Key Element =3

Row operations:

• R1(new)=R1(old) / 3
• R2(new)=R2(old) - R1(new)
• R3(new)=R3(old)
  

Iteration-2		Cj	4	4	0	0	0
B	CB	XB	x1	x2	S1	S2	S3	Min Ratio XB / x2
x1	4	3	1	-0.333	0.333	0	0	---
S2	0	4	0	(1.333)	-0.333	1	0	4 / 1.333=3→
S3	0	25	0	5	0	0	1	255=5
Z=12		Zj	4	-1.333	1.333	0	0
		Zj-Cj	0	-5.333↑	1.333	0	0

Negative minimum Zj-Cj is -5.333 and its column index is 2. So, the entering variable is x2.

The minimum ratio is 3 and its row index is 2. So, the leaving basic variable is S2.

So, the pivot element is 1.333.

Entering =x2, Departing =S2, Key Element =1.333

Row operations:

• R2(new)=R2(old) / 1.333
• R1(new)=R1(old) + 0.333 * R2(new)
• R3(new)=R3(old) - 5 * R2(new)

Iteration-3		Cj	4	4	0	0	0
B	CB	XB	x1	x2	S1	S2	S3
x1	4	4	1	0	0.25	0.25	0
x2	4	3	0	1	-0.25	0.75	0
S3	0	10	0	0	1.25	-3.75	1
Z=28		Zj	4	4	0	4	0
		Zj-Cj	0	0	0	4	0

Since all Zj-Cj ≥ 0, we have reached the optimality condition and the optimal solution is as follows:

x1 = 4
x2 = 3
Max Z = 28

(b)

Note the Zj row entries corresponding to S1, S2, and S3. These are the shadow prices of Constraint-1, 2, and 3 respectively.

So,

the shadow prices are 0, 4, and 0 for Constraint-1, 2, and 3 respectively.

(c)

No. because the shadow price is zero meaning that already there exists slack capacity. Additional resources at any cost are unnecessary.