Consider the following optimization problem:
Maximize: 𝑋1𝑋2
Subject to: 𝑋1 + 𝑋2 = 3; 𝑋1 ≥ 0; 𝑋2 ≥ 0
a. Solve the optimization problem.
b. Compute the shadow prices.
c. Identify the binding and non-binding constraints.
Solve by hand and by Excel Solver, show work for both methods.
Hint: 𝑋1 ≥ 0 can be expressed as −𝑋1 ≤ 0 when adding the slack variable to the inequality
constraint.
Please attach Excel Solver outputs

### Part (a): Solving the Optimization Problem by Hand

**Problem Statement:**

Maximize \( Z = X_1 X_2 \)

**Subject to:**

1. \( X_1 + X_2 = 3 \)
2. \( X_1 \geq 0 \)
3. \( X_2 \geq 0 \)

#### **Step 1: Substitute the constraint into the objective function.**

Given \( X_1 + X_2 = 3 \), we can express \( X_2 \) as:

\[
X_2 = 3 - X_1
\]

Now, substitute \( X_2 = 3 - X_1 \) into the objective function:

\[
Z = X_1(3 - X_1) = 3X_1 - X_1^2
\]

#### **Step 2: Take the derivative of \( Z \) with respect to \( X_1 \) and set it to zero to find the critical points.**

\[
\frac{dZ}{dX_1} = 3 - 2X_1
\]

Set the derivative equal to zero:

\[
3 - 2X_1 = 0
\]

Solve for \( X_1 \):

\[
X_1 = \frac{3}{2} = 1.5
\]

#### **Step 3: Substitute \( X_1 = 1.5 \) back into the constraint to find \( X_2 \).**

\[
X_2 = 3 - X_1 = 3 - 1.5 = 1.5
\]

#### **Step 4: Evaluate the objective function at this point.**

\[
Z = X_1 X_2 = 1.5 \times 1.5 = 2.25
\]

So, the maximum value of \( Z \) is **2.25** when \( X_1 = 1.5 \) and \( X_2 = 1.5 \).

### Part (b): Compute the Shadow Prices

The shadow price (or dual value) represents the change in the objective function value per unit increase in the right-hand side of a constraint.

In this case, the binding constraint is \( X_1 + X_2 = 3 \). 

For linear programming problems like this, the shadow price can be interpreted as the slope of the objective function with respect to the constraint. Since the problem is maximization, the shadow price here is the value of the derivative of the objective function when the constraint is slightly relaxed.

From the earlier derivative:

\[
\frac{dZ}{dX_1} = 3 - 2X_1
\]

At \( X_1 = 1.5 \), this derivative (shadow price) is \( 3 - 2(1.5) = 0 \).

So, the shadow price for the constraint \( X_1 + X_2 = 3 \) is **0**.

### Part (c): Identify the Binding and Non-Binding Constraints

- **Binding Constraint:** \( X_1 + X_2 = 3 \) is binding because the equality holds, and changing the value of the right-hand side of this constraint would affect the optimal solution.

- **Non-Binding Constraints:** \( X_1 \geq 0 \) and \( X_2 \geq 0 \) are non-binding because neither \( X_1 \) nor \( X_2 \) is equal to 0 at the optimal solution (both are equal to 1.5).

### Solving with Excel Solver

To solve this problem using Excel Solver:

1. **Objective Function:** Set \( Z = X_1 \times X_2 \) as the target cell to maximize.
2. **Decision Variables:** Define cells for \( X_1 \) and \( X_2 \).
3. **Constraints:** Add the constraint \( X_1 + X_2 = 3 \) and ensure \( X_1 \geq 0 \) and \( X_2 \geq 0 \).

I’ll generate the Excel Solver setup and outputs for you.

---

**Excel Solver Output:**

*Please upload the Excel file using the "Upload File" option so I can assist you further with Solver outputs and calculations.*

Explore how to maximize the product X1*X2 subject to constraints X1 + X2 = 3, X1 >= 0, X2 >= 0. Learn about shadow prices, binding and non-binding constraints. Ideal for students and professionals interested in linear programming and optimization techniques.

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