2a) Solve this by setting up an appropriate objective function (to optimise – whether minimise or maximise) and constraints, etc. Recalling note 4 and other notes and instructions, with new section in your .pdf file, show all working, decision variables and objective function. (9 marks)

Is it a maximisation or minisation problem?

### Problem Analysis

In this problem, you need to determine how to allocate investments across the available options in order to fund specific payments at different times. You need to maximize returns (interest/yield) while ensuring you meet the payment schedule.

### Type of Problem: **Maximization**

The goal here is to **maximize** the yield from the investments while ensuring that the required payments are made on time. Therefore, this is a maximization problem.

### Decision Variables

Let:
- $ x_{A1}, x_{A2}, \dots, x_{E1} $ be the amount invested in each of the available options (A1, A2, ..., E1).

### Objective Function

The objective function is to maximize the total yield (interest) from all investments.

\[
\text{Maximize } Z = 1.9\% \times x_{A1} + 1.9\% \times x_{A2} + \dots + 10.5\% \times x_{E1}
\]

Where each percentage corresponds to the yield of the respective investment.

### Constraints

1. **Cash Flow Constraints (Meet Payment Requirements):**

For each month when a payment is due, you need to ensure that the funds available from the matured investments are sufficient to cover the payments:

- **Month 2**: $ x_{A1} + x_{A2} + \dots \geq 150,000 $
- **Month 4**: $ x_{B1} + x_{B2} + \dots \geq 200,000 $
- **Month 6**: $ x_{C1} + x_{C4} + \dots \geq 250,000 $
- **Month 8**: $ x_{D1} + x_{D4} + \dots \geq 300,000 $
- **Month 9**: $ x_{E1} \geq 100,000 $

2. **Investment Availability:**
You cannot invest more than what is available. For example:
- $ x_{A1} $ can only be available in month 1.

3. **Total Investment Constraint:**
The total investment should equal $1,000,000:

\[
x_{A1} + x_{A2} + \dots + x_{E1} = 1,000,000
\]

4. **Non-Negativity:**
All decision variables must be non-negative:

\[
x_{A1}, x_{A2}, \dots, x_{E1} \geq 0
\]

### Final Problem Formulation

**Objective Function:**

\[
\text{Maximize } Z = 0.019x_{A1} + 0.019x_{A2} + \dots + 0.105x_{E1}
\]

**Subject to:**

1. $ x_{A1} + x_{A2} + \dots \geq 150,000 $ (for month 2 payment)
2. $ x_{B1} + x_{B2} + \dots \geq 200,000 $ (for month 4 payment)
3. $ x_{C1} + x_{C4} + \dots \geq 250,000 $ (for month 6 payment)
4. $ x_{D1} + x_{D4} + \dots \geq 300,000 $ (for month 8 payment)
5. $ x_{E1} \geq 100,000 $ (for month 9 payment)
6. $ x_{A1} + x_{A2} + \dots + x_{E1} = 1,000,000 $
7. $ x_{A1}, x_{A2}, \dots, x_{E1} \geq 0 $

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Would you like a more detailed solution process or assistance with setting this up for further analysis?

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### 5 Related Questions:
1. How do you determine the appropriate investment amount for each option to meet all constraints?
2. What happens if yields are not maximized while covering the payment schedule?
3. Can all investments be used, or is there an optimal subset?
4. How do investment maturities affect the feasibility of the solution?
5. Are there alternative approaches to solving this maximization problem?

### Tip:
When setting up the objective function and constraints, always double-check that the units (e.g., percentages for yields) are consistent across the equations to avoid errors in optimization models.

Explore how to maximize investment returns while meeting specific payment constraints. Learn about formulating objective functions and applying constraints in optimization problems. Suitable for undergraduate-level studies.

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