Finco must determine how much investment and debtto undertake during the next year. Each dollar invested reduces the NPV of the company by 10¢, and each dollar of debt increases the NPV by 50¢ (due to deductibility of interest payments). Finco can invest at most $1 millionduring the coming year. Debt can be at most 40% of investment. Finco now has $800,000 in cash available. Allinvestment must be paid for from current cash or borrowed money. Set up an LP whose solution will tell Finco how tomaximize its NPV.
x1 = amount invested
x2= amount borrowed
objective function:
Max = -0.10x1+0,5x2
s.t.:
x1 <= 1,000,000
x1 - x2 <= 800,000
x2 <= 0.4x1
max z = .5x1 - .1 x2
s.t constraints
x2< = 1000000
x1<= 400000
x1 + x2 = 800,000
To set up the linear program (LP) for maximizing the NPV, we need to define the decision variables, objective function, and constraints.
Let's define:
x = amount of investment in dollars (in millions)
d = amount of debt in dollars (in millions)
Decision Variables:
x: Investment amount in dollars.
d: Debt amount in dollars.
Objective Function:
Maximize NPV: 0.10x - 0.50d
Constraints:
1. Investment constraint: x <= 1 (as the maximum investment is $1 million).
2. Debt constraint: d <= 0.40x (debt can be at most 40% of the investment).
3. Cash constraint: x + d <= 0.8 (current cash available is $800,000).
Combining all these components, we get the following LP:
Maximize: 0.10x - 0.50d
Subject to:
x <= 1
d <= 0.40x
x + d <= 0.8
This linear program will provide the solution that tells Finco how to maximize its NPV while considering the investment and debt constraints.
To set up a Linear Programming (LP) model for this problem, we need to define the decision variables, objective function, and constraints.
Decision Variables:
Let x be the amount of investment and y be the amount of debt. These are the decision variables that need to be determined.
Objective Function:
The objective is to maximize the Net Present Value (NPV). Given that each dollar invested reduces NPV by 10 cents and each dollar of debt increases NPV by 50 cents, we can express the objective function as follows:
Maximize NPV = -0.10x + 0.50y
Constraints:
1. The total investment and debt cannot exceed the available resources. The total investment is limited to at most $1 million, so we have:
x ≤ 1,000,000
2. Debt can be at most 40% of the investment. This can be defined as:
y ≤ 0.4x
3. The available cash must cover all expenses. The total investment and debt cannot exceed the available cash, i.e., the sum of investment and debt should not exceed $800,000:
x + y ≤ 800,000
4. Non-negativity constraint: Both investment and debt should be non-negative:
x ≥ 0
y ≥ 0
With these variables, objective function, and constraints, we can set up the LP model.
Maximize NPV = -0.10x + 0.50y
subject to:
x ≤ 1,000,000
y ≤ 0.4x
x + y ≤ 800,000
x ≥ 0
y ≥ 0
By solving this LP model, the solution will tell Finco how to maximize its NPV while considering the given constraints.