Graduate student Luke currently has $400 in cash. At the beginning of months 1, 2, 3, and 4, Luke receives revenues from his research assignment, after which he pays bills (see Table). Any money left over is to be invested. If invested for one month, Luke receives the interest rate of 0.1% per month. If invested for two months, he receives an interest rate of 0.5% per month.
Month Revenues ($) Expenses ($)
1 400 600
2 800 500
3 300 500
4 300 250
A. Formulate a linear program to determine an investment strategy that maximizes cash on hand at the beginning of month 5 (or equivalently at the end of Month 4). Please clearly define all terms in your model. (Note, you are not required to solve the model, just formulate it). (15 points)
B. Luke is now able to postpone monthly payments; however, a penalty of 1% is charged on the amount postponed. He can postpone payments by only one month. Formulate a linear program to determine an investment strategy that maximizes cash on hand at the beginning of month 5.
(10 points)
A. To maximize the cash on hand at the beginning of month 5, we can formulate a linear program using the following variables:
Let X1, X2, X3, X4 represent the amounts to be invested at the beginning of months 1, 2, 3, and 4, respectively.
Let Y1, Y2, Y3, Y4 represent the amounts to be postponed at the beginning of months 1, 2, 3, and 4, respectively.
Objective function: Maximize Cash on Hand at the beginning of month 5
Maximize Z = 400 + (0.1% * X1) + (0.5% * X2) + (0.5% * X3) + (0.1% * X4)
Subject to:
X1 + Y1 + 400 - 600 = X2
X2 + Y2 + X1 - 800 - 500 = X3
X3 + Y3 + X2 - 300 - 500 = X4
X4 + Y4 + X3 - 300 - 250 = Z
Constraints:
X1, X2, X3, X4, Y1, Y2, Y3, Y4 >= 0
B. With the ability to postpone payments by one month with a 1% penalty, we can modify the linear program as follows:
Objective function: Maximize Cash on Hand at the beginning of month 5
Maximize Z = 400 + (0.1% * X1) + (0.5% * X2) + (0.5% * X3) + (0.1% * X4)
Subject to:
X1 + Y1 + 400 - 600 - (1% * 600) = X2
X2 + Y2 + X1 - 800 - (1% * 500) = X3
X3 + Y3 + X2 - 300 - (1% * 500) = X4
X4 + Y4 + X3 - 300 - (1% * 250) = Z
Constraints:
X1, X2, X3, X4, Y1, Y2, Y3, Y4 >= 0
In this formulation, the penalty for postponing payments by one month is taken into account by subtracting the penalty percentage from the corresponding expense in the constraints.