Question

Michelle received an inheritance of $95,000, and she is evaluating two speculative investments- the purchase of land and the purchase of cattle. Each investment would be for 1 year. Under "normal" economic conditions, each dollar invested in land will return the principle plus 20% of the principle; each dollar invested in cattle will return the principle plus 30%. However both investments have some risk. If economic conditions worsen, there is an 18% probability theat she will lose everything she invested in land and a 30% probability she would lose everything she invested in cattle. Michelle does not want to lose more than $20,000 (on average). She wants to know how to allocate her investment to maximize the cash value of the investments at the end of 1 year.
Algebraically formulate this linear programming problem.

Answer 1

Variable

x : investment in land

y : investment in cattle

Constraint

1. x and y >= 0

2. x+y <= 95000

3. 0.18x + 0.30 y <= 20000

Objective

Maximize investment+RoI

RoI + investment from land = x * 1.2 * 0.82 + x * 0 * 0.18 = 1.2*0.82 *x

{in 82% cases, the return on land will be 1.2 times the principal and in 18% it will be 0)

similar for cattle, RoI + investment = y* 1.3 *0.70 + y * 0 * 0.30 = 1.3*0.70*y

so maximize 1.2*0.82 *x + 1.3*0.70*y

solution

variable	x	95000
variable	y	0
Constraint1	x and y >= 0
Constraint2	x+y <= 95000	0
Constraint3	0.18x + 0.30 y <= 20000	2900
objective	maximize  1.2*0.82 *x + 1.3*0.70*y	93480

optimization suggest to invest everything in land.