Question

Ashley has earmarked at most $210,000 for investment in three mutual funds: x dollars in a money market fund, y dollars in an international equity fund, and z dollars in a growth-and-income fund. The money market fund has a rate of return of 4%/year, the international equity fund has a rate of return of 6%/year, and the growth-and-income fund has a rate of return of 15%/year. Ashley has stipulated that no more than 30% of her total portfolio should be in the growth-and-income fund and that no more than 55% of her total portfolio should be in the international equity fund. To maximize the return on her investment P (in dollars), how much should Ashley invest in each type of fund?

Maximize		P	=		subject to the constraints
total investments				x+y+z≤210000
growth-and-income percentage				−3x/10​−3y/10​+7z/10​≤0
international equity percentage

Answer 1

x₁ dollars in a money market fund,

x₂ dollars in an international equity fund, and

x₃ dollars in a growth-and-income fund.

.

The money market fund has a rate of return of 4%/year, the international equity fund has a rate of return of 6%/year, and the growth-and-income fund has a rate of return of 15%/year

.

Ashley has earmarked at most $210,000

no more than 30% of her total portfolio should be in the growth-and-income fund

.

no more than 55% of her total portfolio should be in the international equity fund.

.

our system is

subject to

After introducing slack variables

subject to

Iteration-1		Cj	0.04	0.06	0.15	0	0	0
B	CB	XB	x1	x2	x3	S1	S2	S3	MinRatio XB/x3
S1	0	210000	1	1	1	1	0	0	210000/1=210000
S2	0	0	-0.3	-0.3	(0.7)	0	1	0	0/0.7=0→
S3	0	0	-0.55	0.45	-0.55	0	0	1	---
Z=0		Zj	0	0	0	0	0	0
		Zj-Cj	-0.04	-0.06	-0.15↑	0	0	0

Negative minimum Zj-Cj is -0.15 and its column index is 3

Minimum ratio is 0 and its row index is 2.

The pivot element is 0.7.

Entering =x₃, Departing =S₂

Iteration-2		Cj	0.04	0.06	0.15	0	0	0
B	CB	XB	x1	x2	x3	S1	S2	S3	MinRatio XB/x2
S1	0	210000	1.4286	1.4286	0	1	-1.4286	0	210000/1.4286=147000
x3	0.15	0	-0.4286	-0.4286	1	0	1.4286	0	---
S3	0	0	-0.7857	(0.2143)	0	0	0.7857	1	0/0.2143=0→
Z=0		Zj	-0.0643	-0.0643	0.15	0	0.2143	0
		Zj-Cj	-0.1043	-0.1243↑	0	0	0.2143	0

Negative minimum Zj-Cj is -0.1243 and its column index is 2

Minimum ratio is 0 and its row index is 3.

The pivot element is 0.2143.

Entering =x₂, Departing =S₃

Iteration-3		Cj	0.04	0.06	0.15	0	0	0
B	CB	XB	x1	x2	x3	S1	S2	S3	MinRatio XB/x1
S1	0	210000	(6.6667)	0	0	1	-6.6667	-6.6667	210000/6.6667=31500→
x3	0.15	0	-2	0	1	0	3	2	---
x2	0.06	0	-3.6667	1	0	0	3.6667	4.6667	---
Z=0		Zj	-0.52	0.06	0.15	0	0.67	0.58
		Zj-Cj	-0.56↑	0	0	0	0.67	0.58

Negative minimum Zj-Cj is -0.56 and its column index is 1

Minimum ratio is 31500 and its row index is 1.

The pivot element is 6.6667.

Entering =x₁, Departing =S₁,

Iteration-4		Cj	0.04	0.06	0.15	0	0	0
B	CB	XB	x1	x2	x3	S1	S2	S3	MinRatio
x1	0.04	31500	1	0	0	0.15	-1	-1
x3	0.15	63000	0	0	1	0.3	1	0
x2	0.06	115500	0	1	0	0.55	0	1
Z=17640		Zj	0.04	0.06	0.15	0.084	0.11	0.02
		Zj-Cj	0	0	0	0.084	0.11	0.02

Since all

Hence, optimal solution is arrived

.

31500 in a money market fund,

115500 in an international equity fund, and

63000 in a growth-and-income fund.