Question

(Investment): An investor has $150,000 to invest in oil stock, steel stock, and government bonds. The bonds are guaranteed to yield 5%, but the yield for each stock can vary. To protect against major losses, the investor decides that the amount invested in oil stock should not exceed $50,000. The total amount invested in stock CANNOT exceed the amount invested in bonds by more than $25,000.

a) Set up the problem (decision variables, problem constraints, non-negativity constraints).

b) Now form the objective function if oil stock yields 12% and the steel stock yields 9%. How much should be invested in each alternative in order to maximize the return (don't forget the bonds). What is the maximum return?

Answer 1

x₁ = amount invest in oil stock

x₂ = amount invest in steel stock

x₃ = amount invest in government bonds

.

.....................An investor has $150,000 to invest

.

...................the amount invested in oil stock should not exceed $50,000

.

..............The total amount invested in stock CANNOT exceed the amount invested in bonds by more than $25,000.

.

our system is

subject to

.

After introducing slack variables

subject to

.

Iteration-1		Cj	0.12	0.09	0.05	0	0	0
B	CB	XB	x1	x2	x3	S1	S2	S3	MinRatio XB/x1
S1	0	150000	1	1	1	1	0	0	150000/1=150000
S2	0	50000	1	0	0	0	1	0	50000/1=50000
S3	0	25000	(1)	1	-1	0	0	1	25000/1=25000→
Z=0		Zj	0	0	0	0	0	0
		Zj-Cj	-0.12↑	-0.09	-0.05	0	0	0

Negative minimum Zj-Cj is -0.12 and its column index is 1.

Minimum ratio is 25000 and its row index is 3.

The pivot element is 1.

Entering x₁, Departing S₃

.

	Cj	0.12	0.09	0.05	0	0	0
B	CB	XB	x1	x2	x3	S1	S2	S3	MinRatio XB/x3
S1	0	125000	0	0	2	1	0	-1	125000/2=62500
S2	0	25000	0	-1	(1)	0	1	-1	25000/1=25000→
x1	0.12	25000	1	1	-1	0	0	1	---
Z=3000		Zj	0.12	0.12	-0.12	0	0	0.12
		Zj-Cj	0	0.03	-0.17↑	0	0	0.12

Negative minimum Zj-Cj is -0.17 and its column index is 3.

Minimum ratio is 25000 and its row index is 2.

The pivot element is 1.

Entering =x₃, Departing =S₂,

.

Iteration-3		Cj	0.12	0.09	0.05	0	0	0
B	CB	XB	x1	x2	x3	S1	S2	S3	MinRatio XB/x2
S1	0	75000	0	(2)	0	1	-2	1	75000/2=37500→
x3	0.05	25000	0	-1	1	0	1	-1	---
x1	0.12	50000	1	0	0	0	1	0	---
Z=7250		Zj	0.12	-0.05	0.05	0	0.17	-0.05
		Zj-Cj	0	-0.14↑	0	0	0.17	-0.05

Negative minimum Zj-Cj is -0.14 and its column index is 2.

Minimum ratio is 37500 and its row index is 1.

The pivot element is 2.

Entering =x₂, Departing =S₁,
.

Iteration-4		Cj	0.12	0.09	0.05	0	0	0
B	CB	XB	x1	x2	x3	S1	S2	S3	MinRatio
x2	0.09	37500	0	1	0	0.5	-1	0.5
x3	0.05	62500	0	0	1	0.5	0	-0.5
x1	0.12	50000	1	0	0	0	1	0
Z=12500		Zj	0.12	0.09	0.05	0.07	0.03	0.02
		Zj-Cj	0	0	0	0.07	0.03	0.02

Since all  

Hence, optimal solution is arrived

50000 = invest in oil stock

37500= invest in steel stock

62500 = invest in government bonds

.

maximum return is 12500