Investment Net Present Value. Spencer Enterprises is attempting to choose among a series of new investment alternatives. The potential investment alternatives, the net present value of the future stre

Question

Investment Net Present Value. Spencer Enterprises is attempting to choose among a series of new investment alternatives. The potential investment alternatives, the net present value of the future stream of returns, the capital requirements, and the available capital funds over the next three years are summarized as follows. At most one of the warehouse expansion alternatives can be chosen. LO 
2
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4
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5
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7
DATA file Investmentripy
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table
[
[
,
,
Capital Requirements 
(
$
)
]
,
[
Alternative
,
Net Present Value 
(
$
)
,
Year 
1
,
Year 
2
,
Year 
3
]
,
[
Limited warehouse expansion,
4
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0
0
0
,
1
,
0
0
0
,
1
,
0
0
0
,
4
,
0
0
0
]
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[
Extensive warehouse expansion,
6
,
0
0
0
,
2
,
5
0
0
,
3
,
5
0
0
,
3
,
5
0
0
]
,
[
Test market new product,
1
0
,
5
0
0
,
6
,
0
0
0
,
4
,
0
0
0
,
5
,
0
0
0
]
,
[
Advertising campaign,
4
,
0
0
0
,
2
,
0
0
0
,
1
,
5
0
0
,
1
,
8
0
0
]
,
[
Basic research,
8
,
0
0
0
,
5
,
0
0
0
,
1
,
0
0
0
,
4
,
0
0
0
]
,
[
Purchase new equipment,
3
,
0
0
0
,
1
,

VIPSolutions · Accepted Answer

# Investment Net Present Value (NPV) - Integer Programming Model

Let's structure and solve the problem as requested.

## ? **Data Extraction**

| Investment Alternative         | Net Present Value ($) | Year 1 | Year 2 | Year 3 |
|-------------------------------|----------------------|--------|--------|--------|
| Limited warehouse expansion   | 4,000                | 1,000  | 1,000  | 4,000  |
| Extensive warehouse expansion | 6,000                | 2,500  | 3,500  | 3,500  |
| Test market new product       | 10,500               | 6,000  | 4,000  | 5,000  |
| Advertising campaign          | 4,000                | 2,000  | 1,500  | 1,800  |
| Basic research                | 8,000                | 5,000  | 1,000  | 4,000  |
| Purchase new equipment        | 3,000                | 1,000  | 500    | 900    |

**Capital funds available:**
- Year 1: $9,000
- Year 2: $11,000
- Year 3: $12,000

## ? **Decision Variables**

Let:
- $ x_1 $: Limited warehouse expansion (binary)
- $ x_2 $: Extensive warehouse expansion (binary)
- $ x_3 $: Test market new product (binary)
- $ x_4 $: Advertising campaign (binary)
- $ x_5 $: Basic research (binary)
- $ x_6 $: Purchase new equipment (binary)

Each $ x_i $ can be  or 1 (either invest or not).

---

## **a. Integer Programming Model**

### **Objective Function**

Maximize total NPV:
$$
\text{Maximize: } 400x_1 + 600x_2 + 10500x_3 + 400x_4 + 800x_5 + 300x_6
$$

### **Constraints**

#### **Capital Constraints**
- **Year 1:** $ 100x_1 + 250x_2 + 600x_3 + 200x_4 + 500x_5 + 100x_6 \leq 900 $
- **Year 2:** $ 100x_1 + 350x_2 + 400x_3 + 150x_4 + 100x_5 + 500x_6 \leq 11000 $
- **Year 3:** $ 400x_1 + 350x_2 + 500x_3 + 180x_4 + 400x_5 + 900x_6 \leq 12000 $

#### **Warehouse Expansion Constraint**
- At most one warehouse expansion:
    $$
    x_1 + x_2 \leq 1
    $$

#### **Binary Constraints**
- $ x_i \in \{, 1\} $ for all $ i $

---

### **Model Summary**

$$
\begin{align*}
\text{Maximize:} \quad & 400x_1 + 600x_2 + 10500x_3 + 400x_4 + 800x_5 + 300x_6 \
\text{Subject to:} \quad & 100x_1 + 250x_2 + 600x_3 + 200x_4 + 500x_5 + 100x_6 \leq 900 \
& 100x_1 + 350x_2 + 400x_3 + 150x_4 + 100x_5 + 500x_6 \leq 11000 \
& 400x_1 + 350x_2 + 500x_3 + 180x_4 + 400x_5 + 900x_6 \leq 12000 \
& x_1 + x_2 \leq 1 \
& x_i \in \{, 1\} \text{ for } i = 1, \ldots, 6
\end{align*}
$$

---

## **b. Add Dependency Constraint**

**If Advertising campaign ($x_4$) is chosen, Test market new product ($x_3$) must be chosen:**
$$
x_4 \leq x_3
$$

---

## **c. Force Extensive Warehouse Expansion**

**Require: Extensive warehouse expansion must be chosen ($x_2 = 1$).**

---

## **Solving the Model**

Solving this integer programming model requires a solver (e.g., Excel/Google Sheets Solver, Python PuLP, or similar). I will provide a solution approach and likely choices based on the constraints.

---

### **a. Base Case Solution (No Extra Constraints)**

- The main limiting factor is the capital required. The best combination maximizes NPV without breaching any year’s capital limit.
- You must select at most one warehouse expansion.

#### **Manual Selection (Greedy/Heuristic Approach)**

Try high NPV per dollar spent first:

- Try Test market new product ($x_3=1$), NPV=10,500, capital: 6,000/4,000/5,000.
- Remaining year 1: 3,000, year 2: 7,000, year 3: 7,000.

Try Basic research ($x_5=1$), NPV=8,000, capital: 5,000/1,000/4,000.
- Remaining: year 1: 3,000-5,000 (not enough left). So not possible with $x_3=1$.

Try Limited warehouse ($x_1=1$), NPV=4,000, capital: 1,000/1,000/4,000.
- Remaining: year 1: 2,000, year 2: 6,000, year 3: 3,000.

Try Purchase new equipment ($x_6=1$), NPV=3,000, capital: 1,000/500/900.
- Remaining: year 1: 1,000, year 2: 5,500, year 3: 2,100.

Try Advertising campaign ($x_4=1$), NPV=4,000, capital: 2,000/1,500/1,800.
- Year 1: 1,000-2,000 (not enough), so can't add.

So, the best selection in this heuristic is:
- Test market new product ($x_3=1$), Limited warehouse ($x_1=1$), Purchase equipment ($x_6=1$)
- Total NPV: 10,500 + 4,000 + 3,000 = **17,500**

Check if capital constraints are met:
- Year 1: 6,000 + 1,000 + 1,000 = 8,000 ≤ 9,000 ✓
- Year 2: 4,000 + 1,000 + 500 = 5,500 ≤ 11,000 ✓
- Year 3: 5,000 + 4,000 + 900 = 9,900 ≤ 12,000 ✓

**This is a feasible and potentially optimal solution.**

---

### **b. Add Dependency Constraint ($x_4 \leq x_3$)**

If advertising campaign is chosen, test market new product must be chosen.

From above, we did not select the advertising campaign, so the solution remains unchanged at **17,500 NPV**.

---

### **c. Force Extensive Warehouse Expansion ($x_2 = 1$)**

We must set $x_2=1$. Now, try feasible combinations:

- Extensive warehouse ($x_2=1$): NPV=6,000, capital: 2,500/3,500/3,500.
- Remaining: year 1: 6,500, year 2: 7,500, year 3: 8,500.

Try Test market new product ($x_3=1$): 6,000/4,000/5,000.
- Remaining: year 1: 500, year 2: 3,500, year 3: 3,500.

Try Purchase new equipment ($x_6=1$): 1

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